nia 


UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


C.  J.  CLAY  AND  SONS, 
CAMBRIDGE   UNIVERSITY  PEESS   WAREHOUSE, 
AVE   MARIA  LANE. 


£ambrtflse:    DEIGHTON,  BELL,  AND  CO. 

Iripjis:    F.  A.  BROCKHAUS. 
0cb3  gorfe:   MACMILLAN  AND  CO. 


ELEMENTARY   THERMODYNAMICS 


BY 


J.   PARKER,   M.A., 

FELLOW   OF   ST   JOHN'S    COLLEGE,    CAMBRIDGE. 


CAMBRIDGE: 
AT  THE  UNIVERSITY   PRESS. 

1891 

[All  Rights  reserved.] 


(Eambrtlige : 

PRINTED    BY   C.   J.    CLAY,    SI.A.   AND   SONS, 
AT    THE    UNIVERSITY   PRESS. 


POINTED   IN  GREAT  BRITAIN) 


NOTE. 

BEGINNERS  may  omit  the  following  parts : — 

Chapter  I.,  Arts.  20—25. 

Chapter  III.,  Arts.  65,  72—90. 

M         The  word  '  Elementary '  is  used  in  the  title  of  this 
g  book  because  it  does  not  enter  into  the  details  of  elec- 
S  tricity  and  magnetism. 
E 


445346 


CONTENTS. 


CHAPTER   I. 
THE  CONSERVATION  OP  ENERGY 


PAGES 

1-80 


CHAPTER  II. 


ON  PERFECT  GASES 


81-103 


CHAPTER  III. 
CARNOT'S  PRINCIPLE 


.     104-236 


CHAPTER   IV. 
APPLICATIONS  OF  CARNOT'S  PRINCIPLE     , 


.     237-324 


CHAPTER  V. 

THE  THERMODYNAMIC  POTENTIAL 


.     325-343 


CONTEXTS. 


CHAPTER   VI. 

PAGES 

APPLICATIONS  OF  THE  THERMODYNAMIC  POTENTIAL         .    344-37J 


NOTE  A 379-381 

NOTE  B 382-391 

APPENDIX  .  .     393-408 


ELEMENTAKY    THEKMODYNAMICS, 

CHAPTER  I. 

THE   CONSERVATION   OF   ENERGY. 

1.  Units  of  Measurement. — As  it  is  now  becoming 
universal  to  express  all  dynamical  quantities  in  terms  of  a 
very  convenient  set  of  units  based  on  the  Metric,  or 
French,  system  of  weights  and  measures,  we  shall  give  an 
account  of  the  method,  and  also  make  a  comparison  with 
the  less  simple,  but  more  familiar,  English  system. 

In  the  metric  system,  which  was  established  in  France 
by  law  in  1795,  and  is  now  widely  adopted  for  commercial 
purposes,  the  standards  of  length  and  mass  are  the  Metre 
and  the  Kilogramme,  respectively.  The  Metre  is  the 
distance  between  the  ends  of  a  rod  of  platinum  made  by 
Borda,  when  the  temperature  is  that  of  melting  ice :  the 
Kilogramme  is  the  mass  of  a  piece  of  platinum,  also  made 
by  Borda. 

The  subsidiary  measures  of  the  metric  system  are 
formed  as  follows : — 

1  kilometre    =  1000  metres.  1  kilogramme   =  1000  grammes. 

1  hectometre  =   100  metres.  1  hectogramme  =   100  grammes. 

1  decametre  =     10  metres.  1  decagramme  =     10  grammes. 

1  decimetre  =     y\j  metre.  1  decigramme  =     T^  gramme. 

1  centimetre  =  ^  metre.  1  centigramme  =   Tjs  gramme. 

1  millimetre  =Y^JU  metre.  1  milligramme  =  ]7JW  gramme. 
P.  1 


2  ELEMENTARY   THERMODYNAMICS. 

The  Litre  (used  for  liquids)  is  the  same  as  the  cubic 
decimetre. 

The  standards  of  the  metric  system  were  originally 
chosen  so  that  the  metre  should  be  the  ten-millionth  part 
of  the  distance  from  the  pole  to  the  equator,  and  the 
kilogramme  the  mass  of  a  litre  of  distilled  water,  at  4°  C., 
the  temperature  of  maximum  density,  as  nearly  as  could 
be  then  determined. 

In  the  centimetre-gramme-second,  or  C.G.S.,  system  of 
absolute  units,  now  so  generally  used  in  dynamics,  the 
fundamental  units  of  length,  mass,  and  time,  are  chosen 
to  be  the  centimetre,  gramme,  and  mean  solar  second, 
respectively.  The  C.G.s.  absolute  unit  of  force,  called  a 
Dyne,  is  then  defined  to  be  that  force  which,  acting  for 
one  second  on  the  mass  of  a  gramme,  will  generate  a 
velocity  of  one  centimetre  per  second.  The  C.G.S.  absolute 
unit  of  pressure  is  a  pressure  of  one  dyne  per  square 
centimetre. 

In  conjunction  with  these  absolute  units,  we  frequently 
employ  arbitrary  units,  as  the  gramme-weight  for  force, 
and  the  millimetre  of  mercury  and  the  atmo  for  pressures. 

As  the  weight  of  a  given  mass  is  not  quite  the  same 
in  all  parts  of  the  world,  whenever  we  speak  of  a  gramme- 
weight  as  a  measure  of  force,  we  shall  mean  a  force  equal 
to  the  weight  of  a  gramme  at  Paris.  The  acceleration  of 
any  body  falling  freely  at  Paris  under  its  own  weight 
being  980'868  centimetres  per  second,  we  see  that  the 
weight  of  a  gramme  at  Paris  is  980*868  dynes.  To 
determine  the  weight  of  a  gramme  at  any  other  place,  we 
may  use  Clairaut's  formula 

g  =  G  (1  -  -0025659  cos  2X)  (l  -  T32  - 


THE   CONSERVATION   OF   ENERGY.  3 

where  X  is  the  latitude  of  the  place,  z  its  height  above  the 
mean  level  of  the  sea,  r  the  radius  of  the  earth,  g  the 
number  of  dynes  in  a  gramme-weight  at  the  place,  and 
G  the  value  of  g  at  the  mean  level  of  the  sea  in  latitude 
45°. 

The  Atmo  is  denned  to  be  the  pressure  produced  (at 
Paris)  by  a  column  of  mercury  760  millimetres  high,  when 
the  temperature  is  0°  C.,  and  the  top  of  the  mercury 
subjected  to  no  force  but  the  pressure  of  its  own  vapour. 
It  is  found  that  an  atmo  is  equal  to  the  weight  (at  Paris) 
of  1033-279  grammes  per  square  centimetre,  or  to  a 
pressure  of  1013510  dynes  per  square  centimetre. 

The  Atmo,  the  name  of  which  is  due  to  Prof.  J. 
Thomson,  is  about  equal  to  the  average  pressure  of  the 
atmosphere  in  ordinary  places  and  is  chiefly  used  for 
measuring  high  pressures:  the  gramme-weight  is  very- 
convenient  in  estimating  small  forces. 

In  the  English,  or  practical,  system  of  dynamical  units, 
the  fundamental  units  of  length,  mass,  and  time,  are  the 
foot,  the  pound  avoirdupois,  and  the  mean  solar  second, 
respectively.  The  absolute  unit  of  force,  which  Prof.  J. 
Thomson  calls  a  Poundal,  is  that  force  which,  if  it  acted 
for  a  second  on  the  mass  of  a  pound,  would  generate  in  it 
a  velocity  of  one  foot  per  second.  The  accelerating  effect 
of  gravity  being  32'1889  feet  per  second  at  London,  it 
follows  that  the  weight  of  a  pound  at  London  is  32'1889 
poundals.  Hence,  for  rough  purposes,  we  may  consider  a 
poundal  equal  to  the  weight  of  half  an  ounce  in  any  part 
of  the  world. 

We  shall  always  work  in  the  C.G.S.  system  of  units, 
but  it  will  be  easy  to  express  our  results  in  the  English 
practical  method  by  means  of  the  following  data : 

1—2 


4  ELEMENTARY  THERMODYNAMICS. 

1  metre         =39-370432  inches.       1  inch    =  2-54  centimetres  * 
1  centimetre  =     -393704  inch.          1  foot    =30-48  centimetres ,* 
1  sq.  metre   =     1550'03  sq.  in.         1  sq.  in.  =  6 "45  sq.  centimetres  * 
*  Nearly. 

1  litre  =  61-025  cubic  inches  =  l'76l7  pint. 

1  pint  =  '5676  litre. 

1  kilogramme =2 -2046  Ibs.  avoir.  =  15432  grains. 

Hence  we  have,  roughly : — 

Weight  of  1  Ib.  avoir.  =  444900  dynes. 

1  Ib.  per  sq.  inch  =  70  '3  grammes  per  sq.  centimetre  =  69000 

dynes  per  sq.  centimetre. 

1  gramme  per  sq.  centimetre  =  '01 42  Ib.  per  sq.  inch. 
1  atmo  =  14-7  Ibs.  per  sq.  inch. 

2.     The  Conservation  of  Energy.— We  are  now 

prepared  to  explain  the  principle  of  the  Conservation  of 
Energy.  This  is  a  deduction,  by  means  of  experiment  and 
experience,  from  a  more  obvious  result  known  as  the 
principle  of  Work,  which  will  now  be  obtained  as  a  direct 
theoretical  consequence  of  Newton's  three  Laws  of  Motion. 

In  order  to  apply  these  laws  correctly,  it  is  necessary 
to  conceive  the  bodies  which  we  are  considering  to  be 
made  up  of  a  large  number  of  very  small  pieces,  or 
particles,  each  of  which  is  so  small  in  all  its  dimensions 
that  for  all  dynamical  purposes  it  may  be  treated  as  a 
mathematical  point ;  the  relative  motions  of  its  parts  with 
respect  to  one  another  being  negligible  in  comparison  with 
its  motion  of  translation  as  a  whole. 

Before  we  are  able  to  consider  real  bodies  of  finite  size, 
we  must  take  the  case  of  a  single  particle. 

3.  When  a  particle,  which  is  moving  about  in  any 
manner  under  the  action  of  a  force  of  p  dynes,  receives  a 


THE   CONSERVATION   OF   ENERGY.  5 

small  displacement  of  ds  centimetres,  in  a  direction  which 
is  not  at  right  angles  to  the  force,  the  force  is  said  to  do 
Work  on  the  particle,  or  the  particle  is  said  to  do  work 
against  the  force,  and  the  amount  of  work  done  on  the 
particle  is  defined  to  be  the  product  of  the  displacement 
into  the  resolved  part  of  the  force  along  the  displacement. 
If  e  be  the  angle  between  the  positive  directions  of  p  and 
ds,  the  work  done  by  the  force  will  therefore  be  p  cos  e .  dst 
and  may  obviously  be  either  positive  or  negative.  It  is 
also  clear  that  the  work  done  by  the  force  is  equal  to  the 
product  of  the  force  and  the  projection  of  the  displacement 
along  the  force. 

The  absolute  unit  of  work  in  the  c.  G.  s.  system  of  units 
is  called  an  Erg,  and  is  the  work  done  by  a  force  of  one 
dyne  when  it  moves  its  point  of  application  one  centi- 
metre in  its  positive  direction.  The  absolute  unit  of  work 
in  the  English  system  is  the  Foot-poundal,  which  is  the 
work  done  by  a  force  of  one  poundal  when  its  point  of 
application  moves  one  foot  in  the  positive  direction  of  the 
force. 

Work  is  also  reckoned  in  arbitrary  units,  as  the 
gramme-centimetre  and  the  foot-pound.  These  different 
methods  may  easily  be  compared  by  means  of  the  following 

relations : 

1  gramme-centimetre  (at  Paris)  =  980'868  ergs. 
1  foot-pound  =  1356  x  104  ergs,  roughly. 

If  several  forces  act  simultaneously  on  the  particle,  the 
work  which  they  do  is  defined  to  be  the  work  done  by 
their  resultant.  And  since  the  resolved  part  of  the  result- 
ant along  the  displacement  is  equal  to  the  sum  of  the 
resolved  parts  of  the  component  forces  in  the  same 
direction,  it  is  evident  the  work  done  by  the  resultant  is 


6  ELEMENTARY  THERMODYNAMICS. 

equal  to  the  sum  of  the  quantities  of  work  done  by  the 
forces  separately.  Thus  if  R  be  the  sum  of  the  resolved 
parts  of  the  forces  along  the  displacement  ds,  and  dW  the 
work  done  on  the  particle,  we  have  d  W  =  Rds. 

To  put  this  into  analytical  language,  let  (X,  Y,  Z)  be 
the  sums  of  the  components  of  the  forces  which  act  on 
the  particle,  parallel  to  three  fixed  rectangular  axes  (Ox, 
Oy,  Oz),  with  which  the  displacement  ds  makes  angles 
(a,  /3,  7),  respectively.  Then,  by  the  principles  of  statics, 

R  =  X  cos  a  +  Fcos  /3  +  Z  cos  7. 

But  if  (dx,  dy,  dz)  be  the  projections  of  the  displace- 
ment ds  on  the  axes,  we  have,  since  the  axes  are  rect- 
angular, 

dx  =  ds  cos  a,        dy  =  ds  cos  /3,         dz  —  ds  cos  7. 

Hence  dW=  Rds  =  Xdx  +  Ydy  +  Zdz (1). 

If  the  particle  move  from  an  initial  position  A,  whose 
coordinates  are  (>0,  y0,  z0\  to  a  final  position  B,  whose 
coordinates  are  («„  ylt  *,),  the  work  done  on  it  by  the 
forces  may  be  written 

j*dW    or    j\Xdx+ Ydy  +  Zdz), 

where  the  integral  sign  simply  denotes  a  summation, 
without  implying  that  (x,  y,  z)  can  be  taken  as  indepen- 
dent variables. 

4.  Again,  however  complicated  may  be  the  connec- 
tions of  the  given  particle  with  other  particles,  we  may 
always,  for  dynamical  purposes,  treat  it  as  a  perfectly  free 
particle  whose  motions  depend  only  on  the  forces  which 
act  upon  it.  We  may  therefore  take  the  time  as  the 


THE   CONSERVATION  OF   ENERGY.  7 

independent  variable.  Hence,  if  m  be  the  mass  of  the 
particle  in  grammes,  and  (u,  v,  w)  its  velocities  parallel  to 
the  axes  in  centimetres  per  second,  we  have,  by  the  second 
law  of  motion 

du      „  dv     -p.  dw      „ 

m~dt  =  X>     mdt  =  T'     mW  =  Z" 

Thus,  since 

dx  =  udt,        dy  =  vdt,         dz  =  wdt, 
we  obtain 

(du       dv        dw\  ,       „  7       „  7        „, 
U'di+Vdi+Wdi)  +      y  +        ' 

and  therefore  when  the  particle  moves  from  (x0,  y0,  z0)  to 
(#i>  2/i  >  Zi),  we  shall  have 


=  f 
Jo 


But,  since  the  axes  are  rectangular, 

Xdx  +  Tdy  +  Zdz  =  d  W, 

and  if  V  be  the  total  or  resultant  velocity  of  the  particle 
at  any  instant,  we  have  F2  =  u2  +  v2  +  w2,  so  that 


(2). 


Now  the  Kinetic  Energy  of  a  particle  is  denned  to  be 
half  the  product  of  its  mass  and  the  square  of  its  velocity. 
The  last  result  therefore  shows  that  as  the  particle  moves 
about,  the  algebraic  increase  of  its  kinetic  energy  is  equal 
to  the  work  done  upon  it. 

It  is  clear  that  kinetic  energy  is  measured  in  the  same 
way  as  work,  that  is,  in  ergs  (or  foot-poundals). 


8  .ELEMENTARY  THERMODYNAMICS. 

5.  The  forces  which  act  on  the  particle  are  due, 
partly  at  least,  to  the  influence  of  other  particles,  and  as 
these  may  themselves  be  moving  about,  it  is  clear  that 
the  components  (X,  Y,  Z}  will  not  necessarily  depend  on 
the  position  of  the  given  particle  alone.  Hence,  when  the 
particle  moves  from  A  to  B,  the  work  done  upon  it  and 
therefore  also  the  change  in  its  kinetic  energy,  will  not 
generally  depend  solely  on  the  positions  of  A  and  B,  and 
on  the  path  along  which  the  motion  takes  place.  But  if 
(X,  Y,  Z)  depend  only  on  the  position  of  the  given 
particle,  they  will  be  functions  of  its  coordinates  (x,  y,  z), 
and  the  work  done  by  the  forces  will  depend  only  on  the 
two  positions  A,  B,  and  on  the  path  by  which  the  particle 
travels  from  one  to  the  other.  In  this  case,  the  integral 

f  d  W,  or    ^  (Xdx  +  Ydy  +  Zdz), 

Jo  Jo 

falls  under  one  or  other  of  two  heads  between  which  there 
is  a  very  important  distinction,  illustrating  a  point  of 
frequent  occurrence  in  thermodynamics. 

First,  if  (X,  Y,  Z)  satisfy  the  three  relations 
dX  =  dY      dY  _dZ      dZ_dX 

dy      das'     dz~dy'      dx~  ~dz~ (3'' 

then  it  can  be  shown  that  the  expression 

Xdx  +  Ydy  +  Zdz,  or  d  W, 

is  the  complete  differential  of  some  function  of  (x,  y,  z},  so 
that  we  may  write 

W=F(x,y,z)+C, 
where  C  is  an  arbitrary  constant. 

Hence  if  the  particle  move  from  the  position  A  to  the 
position  B,  the  work  done  by  the  forces  takes  the  form 


THE   CONSERVATION   OF  ENERGY. 


9 


and  therefore  when  the  last  position  is  the  same  as  the 
first,  the  work  done  on  the  particle  will  be 


~ 

Now  if  the  function  F  contain  such  terms  as  tan"1  —  , 

xy 

which  may  have  different  values  for  the  same  values  of 
(x,y,z),  the  last  expression  will  not  necessarily  be  zero. 
But  if  we  suppose  F  (x,  y,  z)  to  be  such  that  it  has  a 
single  value  for  each  point  in  space,  in  other  words,  that 
it  is  a  single-valued  function,  then  the  results  just  obtained 
show  that  when  the  particle  moves  from  one  position  to 
another,  the  work  done  by  the  forces  depends  only  on  the 
initial  and  final  positions,  and  is  therefore  the  same  by 
whatever  path  the  particle  may  have  travelled  from  one 
to  the  other  :  also  that  when  the  last  position  coincides 
with  the  first,  the  work  done  is  zero. 

Secondly,  when  the  three  conditions  in  equation  (3) 
are  not  all  satisfied,  the  expression  for  d  W  will  not  be  a 
complete  differential  and  the  integration  will  be  impossible 
until  something  further  is  specified  about  the  path.  This 
will  be  made  clear  by  an  example. 


10  ELEMENTARY  THERMODYNAMICS. 

Let  us  suppose  that  X  =  py,  where  p,  is  a  constant, 
and  that  both  F  and  Z  are  constantly  zero.  Then  we 
have 

d  W  =  pydx. 

If  then  the  curve  PQ  be  the  projection  on  the  plane  of 
(xy)  of  the  path  of  the  particle,  and  PM,  QN  perpendiculars 
from  P  and  Q  on  Ox,  the  work  done  on  the  particle  by  the 
forces  will  be  /*  times  the  area  of  the  figure  PQNM,  which 
will  depend  on  the  form  of  the  curve  PQ  as  well  as  on  the 
positions  of  the  points  P,  Q. 

In  general,  if  a  particle  pass  from  any  position  A  to 
any  other  position  B,  under  the  influence  of  forces  which 
depend  only  on  the  position  of  the  particle  but  do  not 
make  the  expression  Xdx  +  Ydy  +  Zdz,  or  d  W,  a  com- 
plete differential,  the  work  done  by  the  forces  will  depend 
not  only  on  the  two  positions  A,  B,  but  also  on  the  path 
by  which  the  particle  travels  from  A  to  B.  Hence  if  the 
particle  return  from  B  to  A  under  the  influence  of  the 
same  forces  but  by  a  different  path,  the  work  done  in 
the  return  path  will  not  necessarily  be  equal  and  op- 
posite to  that  done  in  the  first  case.  The  total  work 
done  will  therefore  generally  be  different  from  zero. 
Hence  if  the  particle  pass  through  the  same  point 
several  times,  its  velocity  will  generally  be  different 
every  time. 

Again,  even  when  no  function  W  exists  of  which  dW 
is  the  exact  differential,  it  is  still  found  convenient  to 
employ  the  ordinary  notation  of  partial  differential  co- 
efficients. Thus  the  equation 


dW  =  Xdx  +  Ydy  +  Zdz 


THE   CONSERVATION   OF  ENERGY.  11 

gives,  when  dy  and  dz  are  both  zero, 


which  we  shall  write  in  the  form  \—j-  )  =  X.     Similarly 

we  may  write  [-5—]  =  F  and  (—7— )  =Z\  but  we  must 
V  dy  J  V  dz  1 

,       .,    ,   d  fdW\  .  d  (dW\    , 

remember  that  -=-    -= —    is  not  equal  to  -j-    -^—      &c., 
dx  \  dy  )  dy\dxJt 

&c.     In  fact,  we  have  just  seen  that  these  are  the  very 
conditions  that  make  d  W  an  exact  differential. 

6.  A  system  of  real  bodies  is  to  be  considered  as  an 
assemblage  of  a  vast  number  of  particles,  and  its  kinetic 
energy  and  the  work  upon  it  in  any  given  time  are  defined 
to  be  the  sums  of  the  corresponding  quantities  for  the 
individual  particles.  Hence  if  T  be  the  kinetic  energy  of 
the  system  at  any  instant,  T  4-  dT  the  kinetic  energy  at  a 
consecutive  instant,  and  dW  the  work  done  on  the  system 
in  the  interval,  we  shall  have 

dT=dW. 

This  equation  has  been  simplified  by  the  decisive 
discovery  of  the  existence  of  the  Ether.  This  is  a  friction- 
less  substance  or  medium  pervading  all  space  which  is  so 
subtle  that  it  cannot  be  removed  from  a  vessel  in  the 
process  of  forming  a  '  vacuum.'  Not  being  directly  appa- 
rent to  our  senses,  it  is  not  considered  to  form  part  of  our 
material  system.  Of  the  various  functions  which  it  is 
known  to  perform  in  nature,  one  of  the  most  obvious  is 
the  transmission  of  light  and  heat  across  the  vacant  space 
which  separates  the  sun  from  the  earth.  This  was  dis- 
covered by  Roemer  in  1675  to  be  a  gradual  and  not  an 


12  ELEMENTARY   THERMODYNAMICS. 

instantaneous  process,  the  velocity  of  light  in  a  vacuum 
being  about  2998601  kilometres,  or  186326  miles,  per 
second.  The  fact  that  the  presence  of  the  ether  does 
not  sensibly  impede  the  motions  of  the  planets  is  illus- 
trated by  a  well-known  proposition  in  hydrodynamics, 
that  a  body  of  invariable  form  experiences  no  resistance 
in  moving  through  a  frictionless  fluid  with  uniform  velocity 
in  a  straight  line. 

According  to  modern  ideas,  the  forces  which  act  on  any 
particle  consist  of  pushes  and  pulls  due  to  its  direct 
contact  with  other  particles  or  with  the  ether.  Forces 
may  therefore  be  distinguished  as  contact-forces  and 
ether-forces,  respectively,  and  if  dWc  be  the  work  done  on 
the  whole  system  by  the  contact-forces  and  d  We  the  work 
done  by  the  ether-forces,  we  shall  have 
dT=dW=dWc  +  dWe. 

But  when  any  two  particles  are  in  contact,  their  mutual 
contact-forces  are  equal  and  opposite,  and  as  soon  as  the 
particles  separate,  the  forces  both  vanish.  Thus  so  long  as 
the  forces  exist,  their  points  of  application  are  coincident, 
and  therefore  the  quantities  of  work  which  they  do  in  any 
the  same  time  are  equal  and  opposite,  or  the  sum  of  the 
quantities  of  work  is  zero.  Hence,  for  the  whole  system, 
the  work  done  by  the  internal  contact-forces  in  any  given 
time  is  zero,  and  therefore  if  dwc  be  the  work  done  by  the 
external  contact-forces,  we  have 

dwc  =  dWc, 
so  that  the  principle  of  work  takes  the  final  form 

dT=dwc  +  dWe (4). 

1  This  is  Newcomb's  determination,  given  in  Everett's  'Units  and 
Physical  Constants. ' 


THE   CONSERVATION   OF   ENERGY.  13 

7.  The  value  of  the  principle  of  energy  arises  from 
the  fact  that  all  the  ether-forces  in  nature  are  such  as  to 
give  to  dWe  a  very  simple  form.  The  methods  by  which 
this  important  result  has  been  established  are  of  two  kinds. 
First,  there  are  a  number  of  experiments  which  make  it 
appear  extremely  probable,  and  secondly,  whenever  the 
principle  of  energy  has  been  made  use  of  in  the  study  of 
physical  phenomena,  it  has  been  found  to  lead  to  conse- 
quences which  are  in  all  cases  in  exact  agreement  with 
observation ;  indeed,  the  theoretical  calculations  have 
frequently  been  performed  first  and  the  experimental 
verifications  obtained  afterwards. 

If  all  the  different  varieties  of  ether-forces  were 
sufficiently  understood  by  us,  we  should  probably  be 
able  to  give  a  general  theoretical  proof  of  the  principle  of 
Energy.  Unfortunately,  we  know  very  little  about  forces, 
especially  about  those  which  exist  in  the  interior  of  solid 
and  fluid  bodies ;  still  there  are  a  few  forces  which  can  be 
dealt  with  by  theory  alone.  We  shall  therefore  adopt  the 
following  method  of  treatment : — First  of  all,  the  principle 
will  be  proved  for  a  few  ideal  systems  in  which  the  only 
ether-forces  which  are  supposed  to  exist,  or  to  do  work,  are 
identical  with  certain  very  simple  forces  which  are  of 
constant  occurrence  in  real  systems.  After  this,  some 
of  the  peculiarities  of  other  ether-forces  will  be  noticed ; 
by  means  of  which  a  clear  general  conception  of  the 
principle  will  be  gained.  We  are  then  immediately 
led  to  a  very  simple  theoretical  explanation  of  the 
nature  of  heat.  Lastly,  the  experimental  evidence  will 
be  considered,  and  it  will  be  seen  to  be  a  mere  repe- 
tition of  the  theoretical  arguments  as  to  the  nature  of 
heat. 


14  ELEMENTARY   THERMODYNAMICS. 

8.  In  the  first  place,  let  us  suppose  that  there  are  no 
ether-forces  which  do  work.     Then  if  dw  be  the  work  done 
by  the  external  contact-forces,  we  have 

dT=dw, 

and  therefore  when  the  external  work  is  zero,  the  kinetic 
energy  is  constant,  so  that  the  only  effect  of  the  internal 
forces  is  a  transference  of  kinetic  energy  from  one  part  of 
the  system  to  another. 

9.  Next,  suppose   the   only  ether-forces   are  due  to 
gravitation,  so  that,  according  to  Newton's  discovery,  two 
particles  whose  masses  are  m  and  in  grammes  and  whose 
distance  is  r  centimetres,  attract  one  another  with  equal 

and  opposite  forces  of  \  — -j-  dynes,  where  A,  is  a  constant 
number  which  is  easily  determined  from  astronomical  and 

/»     i  o 

other  considerations  to  be  about  y^—  . 

Gravitation  belongs  to  a  class  of  forces  which  are  known 
as  actions  at  a  distance,  because  they  exist  between  par- 
ticles when  they  are  not  in  contact  with  one  another,  and 
even  when  they  are  separated  by  an  ordinary  vacuum,  as 
we  may  see  in  the  case  of  the  attractions  between  the 
heavenly  bodies.  It  was  formerly  supposed  that  actions 
at  a  distance  were  exerted  across  nothing,  but  it  is  now 
believed  that  they  are  due  to  continuous  contact-forces  in 
the  ether. 

To  find  the  work  done  by  the  internal  gravitational 
forces,  we  take  them  in  pairs.  Suppose,  then,  that  P,  Q 
are  the  positions  of  two  particles  of  masses  (m,  m')  when 
their  distance  is  r,  and  let  P',  Q'  be  neighbouring  positions 
such  that  P'Q'  =  r  +  dr.  Drop  the  perpendiculars  PM 


THE   CONSERVATION   OF   ENERGY.  15 

and  QN  from  P  and  Q  on  P'Q'.    Then  if  dwl  be  the  work 

N 


Q 


done  by  the  mutual  attraction  on  the  particle  at  P  and 
dw2  the  work  done  on  the  particle  at  Q,  we  have 

,  ^  mm     m, 

dw1  =  —  X  —  —  .  P  M 


and  therefore 
dwl  -f  dw2  =  -  X         (P'M  +  Q'N)  =  -  X         (P'Q'  -  MN). 


But  if  e  be  the  small  angle  between  the  directions  of 
PQ  and  P'Q', 

MN  =  r  cos  e  =  r, 

small  quantities  of  the  second  order  being  rejected. 
Thus  dwl  +  dw2  =  —  \  —  —  dr, 

and  the  work  done  on  the  system  by  all  the  internal 
gravitational  forces  may  be  written 

.  ,,  mm'  , 

—  \Z  —  —  dr. 
r2 

which  is  obviously  a  perfect  differential.  Hence  when 
the  system  passes  from  a  state  0  to  another  state  P,  the 
work  done  on  it  by  the  internal  forces  will  be  the  same 
for  all  paths  and  may  be  written  Wop,  since  it  depends 


16  ELEMENTARY  THERMODYNAMICS. 

only  on  the  two  states  0,  P.    Equation  (4)  therefore  gives 

TP-T0  =  fFdw0+  I  dwg  +  WOP, 
Jo  Jo 

where  I*  dwc  and  [  dwg  are  the  quantities  of  work  done 

by  the  external  contact  and  gravitational  forces,  and  the 
suffixes  refer  to  the  respective  states.  For  shortness,  this 
result  may  be  written 

Tp-T0-WOP=\Pdw, 

J  o 

and  [  dw  may  be  called  the  external  work  done  on  the 

system.  If  the  state  0  be  fixed,  the  term  -WOP  will  only 
vary  with  the  state  P,  and  may  therefore  be  written  Vp : 

hence  TP+VP-T0=!  dw    (5). 

J  0 

It  thus  appears  that  if  no  external  work  be  done  on  the 
system,  T+V  remains  constant,  so  that  T  can  only  in- 
crease at  the  expense  of  V,  and  vice  versa.  For  this 
reason  V  is  regarded  as  a  second  kind  of  energy,  and  has 
received  the  appropriate  name  of  Potential  Energy  from 
Rankine,  to  indicate  that  it  is  convertible  into  kinetic 
energy.  Its  value  in  any  state  P  is  equal  to  the  work 
which  the  internal  forces  do  on  the  system  as  it  rettirns 
in  any  manner  from  the  state  P  to  any  other  state  in 
which  the  relative  positions  of  the  particles  are  the  same 
as  in  the  standard  fixed  state  0,  so  that  in  the  standard 
state  the  potential  energy  is  zero. 

It  was  formerly  thought  that  kinetic  energy  was 
actually  created  when  it  increased  at  the  expense  of  the 
potential  energy  in  the  case  of  a  system  under  the  action 


THE  CONSERVATION   OF   ENERGY.  17 

of  no  external  forces,  but  it  is  now  supposed  to  be  merely 
transferred  from  the  ether. 

The  sum  of  the  kinetic  and  potential  energies  is  called 
the  '  Internal  Energy,'  or,  more  shortly,  the  '  Energy  '  of 
the  system,  and  is  usually  written  U1,  so  that  equation 
(5)  becomes 

UP-U0=fdw  ..................  (6). 

J  0 

If  we  take  a  different  state  0'  for  the  standard  fixed 
state  and  denote  the  new  value  of  UP  by  IT  p,  we  have 


whence  LTP-UP=  WOP  -  W0-P. 

Now  since  the  work  done  by  the  internal  forces  is  the 
same  for  all  paths,  we  may  suppose  the  path  O'P  to  pass 
through  0.  Consequently 


and  therefore  U'  P  -UP=-  W0'0, 

which  is  independent  of  the  state  P. 
Hence,  since  it  is  clear  that  the  energy  cannot  have 
two  different  values  corresponding  to  the  same  state  of 
the  system,  it  follows  that  U  is  a  single-valued  function 
of  the  independent  variables  which  define  the  state  of 
the  system,  together  with  an  arbitrary  additive  constant 
depending  only  on  the  choice  of  the  standard  state. 

1  Sometimes,  in  English  books,  the  letter  E  is  used  to  denote 
internal  energy.  The  notation  adopted  in  the  text  appears  to  be 
preferable,  because  E  is  required  in  Electricity  to  denote  '  electro- 
motive force.' 

p.  2 


18  ELEMENTARY  THERMODYNAMICS. 

Thus  if  (as,  y,  z, )  be  the  independent  variables,  we 

have 


where  G  is  an  arbitrary  constant. 

If  we  take  the  difference  between  the  values  of  U  in 
two  different  states  of  the  system,  the  arbitrary  constant 
will  not  appear  in  the  result,  which  is  therefore  perfectly 
definite. 

When  the  system  consists  of  a  number  of  separate 
bodies,  its  energy  depends  not  only  on  the  state  of  each 
of  its  parts  but  also  on  their  relative  positions  with  respect 
to  one  another.  The  energy  which  depends  on  the  relative 
positions  of  the  bodies  is  called  their  '  mutual '  potential 
energy,  or,  shortly,  their  '  mutual  energy,'  and  is  evidently 
the  excess  of  the  energy  of  the  whole  system  over  the 
sum  of  the  internal  energies  of  the  bodies  of  which  it  is 
composed. 

10.  Again,  let  us  suppose  our  system  to  be  influenced 
by  radiation  as  well  as  by  contact-forces  and  gravitation. 
According  to  modern  theories,  radiation  consists  of  spherical 
waves  of  motion  in  the  ether  which  are  excited  by  the 
irregular  vibrations  of  the  smallest  parts,  or  '  atoms1/  of 
matter,  somewhat  after  the  manner  of  the  circular  waves 
which  may  be  produced  by  dropping  a  stone  into  still 
water.  The  reasoning  on  which  this  conclusion  is  based 
involves  optical  principles  which  cannot  be  discussed  in 
this  book,  but  it  will  become  evident  later  on  that  all 
bodies  are  in  a  state  of  vibration.  Thus  a  wave  of  radia- 

1  An  'atom'  is  a  chemical  reality,  a  particle  a  pure  mathematical 
conception  for  the  purpose  of  calculation.  An  atom  generally  contains 
an  infinite  number  of  particles. 


THE  CONSERVATION  OF  ENERGY.          19 

tion  possesses  energy,  but  its  energy  differs  from  the 
potential  energy  of  gravitation  in  not  being  bound  to 
material  bodies  and  carried  about  with  them. 

When  a  wave  of  radiation  falls  on  a  material  system, 
it  will  affect  both  the  potential  and  kinetic  energy  of  the 
system;  but  the  effect  on  the  potential  energy  is  generally 
exceedingly  minute.  Hence,  to  bring  the  subject  of  radia- 
tion within  the  scope  of  this  book,  we  may  assume  that 
the  effect  of  radiation  on  a  system  is  produced  entirely  by 
the  imperceptible  impacts  of  the  waves  on  the  material 
particles  of  the  system1.  If  dwr  be  the  work  done  on 
the  system  by  these  forces,  which  may,  for  shortness,  be 
called  radiation-forces2,  equation  (4)  becomes 

TP-T0=  P  (dwc  +  dwg  +  dwr)  +  WOP. 

•>  0 

Putting  dw  for  dwc  +  dwg  +  dwr  and  supposing  the  state 
0  to  be  fixed,  this  result  may  be  written  in  any  of  the 
forms 

T  +  VP-T0=!P  di 

Jo 

UP-U0=\P  dw 

Jo 

dU=dw 

The  energy  which  is  taken  from  the  system  by  gravita- 
tion or  radiation  passes  into  the  ether,  but  the  energy 

1  That  is,  when  a  body  is  absorbing  radiation,  the  energy  so  gained 
is  equal  to  the  work  done  on  the  material  particles  of  the  body  by  the 
incident  waves;  when  a  body  is  emitting  radiation,  the  energy  so  lost 
is  equal  to  the  work  done  on  the  ether  by  the  vibrating  particles  in 
starting  the  waves. 

2  As  will  be  seen  later,  the  particles  of  a  body  vibrate  in  such  different 
directions,  that  the  resultant  of  the  radiation-forces  which  act  on  them 
will  generally  be  quite  imperceptible. 

2—2 


20  ELEMENTARY  THERMODYNAMICS. 

which  is  lost  owing  to  contact-forces  passes  directly  into 
some  other  system.  Hence,  when  the  external  forces 
which  act  on  a  system  are  all  contact-forces,  the  external 
work  is  frequently  referred  to  as  the  work  done  on  the 
system  by  external  bodies. 

Radiation-forces  are,  of  course,  far  too  small  to  be 
detected  by  instrumental  means;  nevertheless  the  work 

I    can  be  found.     For  if  the  change  from  the  state  0  to 

Jo 

the  state  P  be  effected  by  means  of  a  measured  quantity 
of  work,  W,  done  by  contact-forces  and  gravitation  alone, 
we  have 


The  simple  fact  we  have  just  obtained  for  our  ideal 
systems,  that  the  energy  U  increases  by  the  amount  of 
external  work  done  on  the  system,  or  that  dw  is  a  perfect 
differential,  is  the  great  principle  of  the  Conservation  of 
Energy.  It  appears  to  be  true  for  all  systems  found  in 
nature,  and  in  consequence  all  natural  forces  are  said  to 
be  Conservative. 

11.  The  principal  ether-forces  in  nature  which  do 
work,  in  addition  to  gravitation  and  radiation-forces,  are 
those  which  give  rise  to  chemical,  physical,  electric,  and 
magnetic  actions. 

When  a  chemical  or  physical  process  takes  place  in  a 
system,  there  is  a  large  amount  of  work  done  by  the  ether- 
forces  in  the  interior  of  the  system  ;  but  it  is  found  that 
no  consequent  effect  is  produced  where  the  process  does 
not  actually  take  place,  except  by  contact-forces  and 
radiation.  Hence,  if  there  are  no  electric  or  magnetic 


THE   CONSERVATION   OF  ENERGY.  21 

actions,  our  system  is  necessarily  so  chosen  that  the  only 
external  forces  are  contact-forces,  gravitation,  and  radia- 
tion-forces. It  is  then  found,  by  experiment  and  experience, 
that  the  system  satisfies  a  relation  of  the  form 


or  dU=dw, 

where  U  is  a  single- valued  function  of  the  independent 
variables  which  define  the  state  of  the  system  together 
with  an  arbitrary  additive  constant  depending  only  on 
the  choice  of  the  standard  fixed  state  0.  The  potential 
energy,  it  must  be  remembered,  is  not  the  same  as  if 
gravitation  were  the  only  internal  force. 

Electric  and  magnetic  forces  are  the  only  ether-forces 
besides  gravitation  about  which  much  is  known,  for  the 
simple  reason  that  they  are  the  only  other  ether-forces 
which  have  been  measured.  One  of  the  chief  peculiarities 
in  which  they  differ  from  gravitation  is  that  the  properties 
of  exerting  actions  at  a  distance  may  be  transferred  from 
one  particle  to  another.  In  consequence  of  this,  it  is  found 
that  a  system  may  acquire  such  an  electric  condition  by 
contact  with  external  bodies  that  it  is  impossible  to  bring 
it  into  the  standard  state  until  the  new  electric  properties 
are  given  back.  We  are  then  to  regard  the  system  as  a 
new  system  and  to  choose  a  state  into  which  the  system 
can  be  brought  as  a  new  standard  state  from  which  its 
potential  energy  may  be  reckoned.  The  difficulty  may 
be  obviated  by  extending  our  system  so  as  to  include  the 
bodies  from  which  these  electric  properties  have  been 
obtained.  If  our  system  be  so  chosen,  and  if  there  be  no 
external  electric  or  magnetic  forces,  then  it  is  found, 


22  ELEMENTARY   THERMODYNAMICS. 

however    complicated   may  be   the   chemical   or   electric 
actions  in  its  interior,  that  it  satisfies  the  relation 


o 

or  d  U  =  dw. 

In  applying  the  principle  of  energy  to  an  electrified  or 
magnetized  system,  there  is  a  special  consideration  to  be 
attended  to.  Thus  let  P,  Q  be  two  different  states  of  the 
same  system  which  can  be  compared  with  the  same  stan- 
dard state  or  origin.  Suppose  also  that  the  kinetic  energies 
of  the  particles  are  the  same  in  both  cases,  but  that  the 
potential  energy  is  greater  in  the  state  Q  than  in  the  state 
P.  Then  it  is  found  that  when  there  are  other  electrified 
or  magnetized  bodies  in  the  neighbourhood,  the  system 
may  be  brought  from  the  state  P  to  the  state  Q  without 
doing  external  work.  But  though  the  external  forces  do 
no  work  on  the  material  part  of  the  system,  they  do  work 
on  that  part  of  the  ether  which,  according  to  one  of  Fres- 
nel's  great  discoveries,  is  inseparably  connected  with  the 
system.  In  fact,  it  appears  from  works  on  Electricity, 
that,  under  these  circumstances,  the  bound-ether  may  be 
regarded  as  a  spring  which  can  be  bent  independently  of 
the  material  part  of  the  system.  If  then  dll  be  the  in- 
crease of  the  energy  in  any  change  of  state  and  dw  the 
external  work  done  on  the  material  particles  of  the  system, 
we  must  write 

dU=dw  +  de, 

where  de  may  be  called  the  '  electric  work'  done  on  the 
system. 

Again,  the  system  may  be  acquiring  electric  properties 
in  one  part  of  its  mass  and  simultaneously  losing  in  another 
in  such  a  way  that  the  potential  energy  may  always  be 


THE  CONSERVATION   OF   ENERGY.  23 

reckoned  from  the  same  standard  state  or  origin.  When, 
as  in  the  case  of  electric  currents,  the  properties  which  the 
system  gains  bring  with  them  more  potential  energy  than 
those  which  it  loses  take  away,  the  increase  of  energy  will 
be  different  from  the  external  work  done  on  the  material 
part  of  the  system,  and  we  must  again  write 
dU—dw  +  de. 

From  this  it  is  evident  that  electric  and  magnetic 
actions  occupy  a  place  between  gravitation,  on  the  one 
hand,  and  radiant,  or  free  etherial,  energy,  on  the  other. 

As  we  do  not  intend  to  enter  into  the  details  of  electric 
and  magnetic  forces  in  this  work,  we  shall  always  suppose 
our  system  so  chosen  that  the  only  external  forces  are 
contact-forces,  gravitation,  and  radiation-forces,  except 
when  it  is  expressly  stated  otherwise.  The  principle 
of  energy  may  then  be  stated  in  the  simple  form 


or  dU=dw, 

so  that  dw  is  a  complete  differential. 

It  is  obvious  that  the  only  effect  of  the  internal  ether- 
forces  is  a  transformation  of  kinetic  into  an  equal  amount 
of  potential,  energy,  and  vice  versa. 

Since  the  principle  of  energy  does  not  require  us  to 
consider  the  internal  forces,  we  shall  frequently  drop  the 
adjective  in  referring  to  the  external  forces  or  work.  The 
energy  of  a  system  may  then  be  described  as  its  capacity 
for  doing  work,  and  positive  work  may  be  regarded  as  the 
passage  of  energy  into  the  system. 

We  now  proceed  to  distinguish  kinetic  energy  and 
work  into  their  visible  and  invisible  parts,  so  as  to 
explain  the  phenomena  of  Heat. 


24  ELEMENTARY   THERMODYNAMICS. 

12.  A  very  little  observation  enables  us  to  perceive 
a  general  tendency  in  nature  for  all  solid  bodies,  or  for 
liquids  and  gases  contained  in  vessels,  to  assume  a  sen- 
sibly invariable  form  and  internal  condition,  except  when 
they  are  prevented  by  external  causes  (including  as  such 
the  radiation  of  energy  into  external  space).  Thus  if  we 
strike  a  bell  or  other  body,  vibrations  are  produced 
which  are  frequently  visible  to  the  eye,  but  always 
disappear  from  sight  more  or  less  rapidly.  Indeed, 
after  a  sufficient  time  has  elapsed,  the  most  powerful 
microscope  fails  to  detect  any  vibrations  in  the  body. 
Again,  if  two  moving  bodies  collide,  they  may  be  even- 
tually brought  nearly  to  a  state  of  apparent  rest  by 
the  collision,  so  that  there  will  seem  to  be  a  considerable 
loss  of  kinetic  energy  without  a  corresponding  increase  of 
potential  energy.  And  since  many  cases  occur  in  which 
we  naturally  suppose  very  little  energy  to  be  lost  from 
the  two  colliding  bodies  before  the  vibrations  subside,  we 
are  led  to  conjecture  that  kinetic  energy  may  exist  in  an 
invisible  as  well  as  in  a  visible  and  palpable  form.  It  will 
be  seen  hereafter  that  there  are  experiments  which  raise 
this  suspicion  to  a  certainty.  We  therefore  define  the 
Mechanical  Kinetic  Energy  of  a  system  of  bodies  to  be 
the  kinetic  energy  that  the  system  would  have  if  its 
motions  were  the  same  as  they  appear  to  be,  or,  more 
exactly : — 

If  we  divide  a  material  system  into  a  large  number  of 
parts,  and  then  multiply  the  mass  of  each  by  the  square 
of  the  velocity  of  its  centre  of  mass  and  take  the  sum,  the 
result  obtained  will  approach  a  limit  as  the  number  of 
parts  is  continually  increased;  but  after  this  limit  is 
practically  reached,  it  may  begin  to  diverge  from  it 


THE   CONSERVATION  OF   ENERGY.  25 

and  finally  arrive  at  another  limit,  which  is  the  true 
kinetic  energy  of  the  system.  When  there  are  two  or 
more  limits,  the  first  of  the  improper  limits  is  defined 
to  be  the  Mechanical  Kinetic  Energy  of  the  system, 
and  the  excess  of  the  true  limit  over  this,  the  Non- 
mechanical  Kinetic  Energy.  Since  it  will  appear  that 
kinetic  energy  is  always  present  in  the  invisible  form, 
it  is  clear  that  if  there  is  only  one  limit  in  the  above 
process,  the  mechanical  kinetic  energy  is  zero  and  the 
system  in  a  state  of  mechanical  rest. 

It  can  easily  be  shown  that  the  mechanical  kinetic 
energy  is  always  less  than  the  total  kinetic  energy, 
and,  consequently,  the  non-mechanical  kinetic  energy 
always  positive.  Thus  let  M  be  the  mass  of  one  of  the 
parts  into  which  the  system  has  just  been  divided,  (u,  v,  w) 
the  velocities  of  its  centre  of  mass  parallel  to  three  rect- 
angular axes,  (mjraaWj...)  the  masses  of  its  ultimate 
particles  and  (u^w^,  (u^w^,...  their  velocities  parallel 
to  the  same  axes.  Then  we  have,  by  the  conservation 
of  linear  momentum, 

mjUi  +  ra2w2  +  m3U3  +  ......  =  Mu, 

whence 


Hence  M2u2  cannot  be  greater  than 

m?u?  +  w22M22  +  ra32w32  +  ......  +  m^m*  (u?  +  u/}  +..., 

or  than 


Thus  %Mu2  cannot  be  greater  than  ^Sm^2  ;  and  similarly 
for  the  velocities  parallel  to  the  other  axes  ;  which  proves 
the  proposition. 

The   non-mechanical   kinetic   energy  of  a  system   is 


26  ELEMENTARY   THERMODYNAMICS. 

supposed   to   be   that   part   of  its   energy  on  which   its 
thermal  properties  depend. 

It  should  be  noticed  that  in  the  case  of  elastic  bodies, 
like  iron  or  compressed  gases,  the  capacity  of  acquiring 
mechanical  kinetic  energy  without  the  assistance  of  ex- 
ternal forces — a  property  which  will  hereafter  be  defined 
as  mechanical  potential  energy — is  not  necessarily  due, 
like  true  potential  energy,  to  the  existence  of  internal 
ether-forces. 

13.  In  consequence  of  non-mechanical  kinetic  energy, 
work  may  be  done  by  contact-forces  in  an  invisible  as  well 
as  in  a  visible  manner.  Suppose,  for  example,  that  any 
two  bodies  A,  B  are  in  contact  and  that  the  surface  of 
contact  is  apparently  at  rest.  Then  no  visible  work  is 
done  by  the  contact-forces;  but  if  the  surface  particles 
of  A  and  B  remain  in  contact  through  a  much  greater 
distance  when  the  direction  of  their  vibratory  motion 
is  from  A  towards  B  than  when  it  is  in  the  contrary 
direction,  the  total  work  of  the  contact-forces  done  by 
A  upon  B  may  be  considerable.  We  therefore  define 
the  Mechanical  Work  done  upon  any  system  of  bodies  by 
the  external  contact-forces  to  be  the  work  that  they  would 
actually  do  if  the  motions  of  the  surfaces  of  contact  were 
the  same  as  they  appear  to  be,  or,  more  exactly : — 

If,  in  order  to  find  the  work  done  on  any  system  in 
any  small  change  of  state  by  the  contact-forces  due  to 
external  bodies,  we  divide  the  surfaces  of  contact  into  a 
large  number  of  parts,  and  then  multiply  the  displace- 
ment of  the  centre  of  each  of  these  small  areas  by  the 
resolved  part  in  the  direction  of  the  displacement  of  the 
force  which  acts  upon  it  and  take  the  sum,  the  result 
obtained  will  approach  a  limit  as  the  number  of  parts  is 


THE   CONSERVATION   OF    ENERGY.  27 

continually  increased;  but  after  this  limit  is  practically 
reached,  it  may  begin  to  diverge  from  it  and  finally  arrive 
at  another  limit,  which  is  the  true  work  done  by  the 
contact-forces  on  the  system.  When  there  are  two  or 
more  limits,  the  first  of  the  improper  limits  is  defined  to 
be  the  Mechanical  Work,  and  the  excess  of  the  true  limit 
over  this,  the  Non-mechanical  Work.  If  there  is  only 
one  limit,  either  the  mechanical  or  the  non-mechanical, 
work  is  zero. 

Since  every  force  which  acts  upon  the  system  produces 
a  change  in  some,  at  least,  of  the  independent  variables 
(x,  y,  z,...)  which  define  the  state  of  the  system,  it  is 
evident  that  both  the  mechanical  and  the  non-mechanical 
work  done  on  the  system  in  any  small  change  of  state 
are  functions  of  the  independent  variables  and  of  their 
differentials. 

It  appears  from  experiment  that  the  non-mechanical 
work  done  by  contact-forces  is  what  we  understand  when 
we  speak  of  the  conduction  of  heat. 

In  the  case  of  gravitation,  the  force  between  any  two 
particles  depends  only  on  their  distance  and  therefore 
cannot  change  abruptly,  like  a  contact-force.  From  this 
it  can  be  shown  that  the  whole  of  the  work  done  on  any 
system  by  the  external  gravitational  forces  is  mechanical 
work,  and  that,  in  calculating  it,  we  need  take  no  account 
of  the  non-mechanical  motions.  This  result,  joined  to  the 
fact  that  an  enormous  amount  of  'heat'  is  transmitted 
from  the  sun  to  the  earth,  affords  a  strong  argument  for 
the  existence  of  the  ether. 

Thus,  if  dU  be  the  increase  of  the  energy  of  any 
system  in  any  small  change  of  state,  and  dw  the  total 
work  done  upon  it,  consisting  of  a  quantity  dW  of 


28  ELEMENTARY  THERMODYNAMICS. 

mechanical  work  and  a  quantity  dQ  of  non -mechanical 
work,  due  either  to  the  conduction  of  heat  or  to  radiation, 

we  have 

dU=dw  =  dW  +  dQ (8), 

from  which  we  draw  the  important  conclusion  that  dW 
and  dQ  will  either  both  simultaneously  be,  or  both  not  be, 
complete  differentials  of  functions  of  the  independent 
variables. 

14.  If  the  surface  of  the  smoothest  body  be  examined 
by  a  powerful  microscope,  it  is  found  to  be  so  irregular 
that  it  may  be  said  to  be  covered  with  a  great  number  of 
small  projecting  teeth.  Besides  the  irregularities  revealed 
by  the  microscope,  there  are  probably  a  vast  number  too 
small  to  be  detected.  Hence,  if  any  two  bodies,  A,  B  be 
pressed  together,  their  surfaces  of  contact  will  sink  into 
one  another,  and  if  we  attempt  to  move  one  body  over 
the  other,  we  shall  experience  a  resistance  in  addition  to 
the  external  forces.  This  resistance  is  at  once  recognised 
as  Friction,  and  by  supposing  the  common  surface  of  A,  B 
to  be  plane,  and  taking  account  of  the  vast  number  of 
teeth  found  even  on  the  smallest  area,  we  may  easily 
obtain  some  of  its  chief  '  Laws,'  thus : — 

I.  Friction  acts  on  each  body  in  a  direction  opposite 
to  that  in  which  its  relative  motion  takes  place,  or  merely 
tends  to  take  place. 

II.  No  more  friction  can  ever  be  called  into  play 
than  is  just  sufficient  to  prevent  relative  motion;   but 
since  the  amount  of  friction  cannot  be  infinite,  it  is  clear 
that  the  frictional  resistance  will  be  unable  to  prevent 
relative  motion  when  the  force  which  tends  to  produce  it 
is   large  enough.     Hence,  as  the   force  which   tends   to 


THE   CONSERVATION   OF  ENERGY.  29 

cause  relative  motion  continually  increases  from  zero, 
friction  will  continually  increase  with  it,  at  least  until  the 
common  surfaces  begin  to  slip.  This  is  usually  expressed 
by  saying  that  if  friction  can  prevent  relative  motion,  it 
will.  The  amount  of  friction  called  into  play  between 
any  two  given  bodies  by  a  given  normal  pressure  when 
slipping  is  about  to  take  place,  is  called  the  'limiting' 
friction  for  that  normal  pressure. 

III.  Let  C,  D  be  two  other  bodies  in  contact  whose 
natures  and  conditions  are  the  same  as  those  of  A,  B, 
respectively,  and  suppose  the  total  normal  pressure 
between  C,  D  to  be  n  times  as  great  as  between  A,  B. 
Then  if  the  common  area  of  C,  D  be  n  times  the  common 
area  of  A,  B,  and  if  the  normal  pressures  be  evenly 
distributed  in  both  cases,  it  is  clear  that,  for  these  normal 
pressures,  the  limiting  friction  between  C,  D  will  be 
n  times  as  great  as  the  limiting  friction  between  A,  B. 
From  experiment  it  further  appears,  at  least  very  approxi- 
mately, that  when  the  total  normal  pressures  are  as  n  to  1, 
the  limiting  amounts  of  friction  will  be  in  this  ratio 
whatever  be  the  proportion  of  the  common  areas.  Hence, 
if  R  be  the  total  normal  pressure  between  any  two  given 
bodies  whose  common  surface  is  plane,  the  limiting 
friction  between  them  will  be  pR,  where  /j,  is  a  constant 
number  (called  the  coefficient  of  friction)  depending  on 
the  natures  and  states  of  the  two  bodies,  but  independent 
both  of  the  total  normal  pressure  and  of  the  area  of  the 
surface  of  contact. 

15.  On  account  of  friction,  an  expenditure  of  me- 
chanical work  is  always  necessary  in  order  to  cause  two 
bodies  to  slide  over  one  another.  If,  as  often  happens, 


30  ELEMENTARY   THERMODYNAMICS. 

the  two  bodies  possess  no  appreciable  potential  energy,  or, 
at  least,  only  a  constant  amount,  and  if  the  mechanical 
kinetic  energy  be  zero  both  before  and  after  slipping,  the 
only  effect  on  the  two  bodies  of  the  mechanical  work  done 
on  them  will  be  an  increase  in  their  non-mechanical 
kinetic  energy.  Now  it  is  evident  that  exactly  the  same 
effect  might  have  been  produced  by  doing  an  equal 
amount  of  non-mechanical  work.  Thus  the  same  change 
of  state  may  be  brought  about  in  the  two  given  bodies  by 
means  of  mechanical  work  alone,  or  by  means  of  non- 
mechanical  work  alone,  or  partly  by  means  of  mechanical 
and  partly  by  means  of  non-mechanical,  work,  in  any 
given  ratio.  Hence,  on  account  of  friction,  when  any 
system  experiences  a  change  of  state,  the  quantities  of 
mechanical  and  of  non- mechanical  work  done  upon  it  will 
generally  depend  on  the  way  in  which  the  change  of  state 
takes  place  as  well  as  on  the  initial  and  final  states 
themselves.  In  other  words,  dW  and  dQ,  though  they 
depend  only  on  the  independent  variables  which  define 
the  state  of  the  system,  and  on  their  differentials,  will  not 
generally  be  perfect  differentials  of  functions  of  the 
independent  variables.  This  is  one  of  the  most  important 
results  in  the  whole  science  of  energy. 

When  the  only  forces  which  do  work  between  two 
bodies  A,  B  are  contact-forces,  let  the  total  work  done  by 
A  on  B  consist  of  a  quantity  of  mechanical  work  Fx  and 
a  quantity  of  non-mechanical  work  Q1 ;  also  let  W,,  Qz  be 
the  corresponding  quantities  of  work  done  by  B  on  A. 
Then  we  have 


so  that  Q1  and  Q2  cannot  be  equal  and  opposite  unless  TFj 
and  Wz  be  so  too.     This  requires  that  the  surfaces  of  the 


THE   CONSERVATION   OF   ENERGY.  31 

two  bodies  should  not  slip  over  one  another.  For  ex- 
ample, if  the  surface  of  A  be  at  rest  while  that  of  B  slides, 
we  shall  have  W^  negative,  equal  to  —  W,  say,  and  Wz  zero. 
Hence  Q1  +  Q*-W=Q, 

so  that  Q!  +  Q2  is  n°t  zero. 

16.  In  one  special  case,  of  frequent  occurrence,  the 
value  of  d  W  takes  a  very  simple  form.  If,  for  example, 
a  body  is  exposed  to  the  air,  it  will  be  acted  on  by  only 
three  external  forces  which  do  mechanical  work — the 
pressure  of  the  air,  the  attraction  of  the  earth,  and  the 
reactions  of  the  fluid  or  solid  bodies  with  which  it  is  in 
contact.  If  the  centre  of  mass  of  the  body  move  neither 
up  nor  down,  and  if  the  reactions  of  the  contiguous 
objects  do  no  work,  the  two  latter  forces  need  not  be 
considered.  The  only  force  with  which  we  are  concerned 
is  therefore  a  uniform  normal  pressure,  which  will  gene- 
rally also  be  constant,  on  certain  parts  of  the  surface. 

The  uniform  normal  pressure  on  the  surface  being 
denoted  by  p,  and  the  volume  by  v,  the  pressure  on  a 

dn 


small  area  da.  of  the  surface  will  be  pda,  and  hence  when 
this  small  area  is  forced  out  a  small  normal  distance  dn 
by  the  expansion  of  the  body,  the  mechanical  work  done 
upon  it  by  the  pressure  will  be 

—  pdadn. 

Since  the  pressure  is  uniform,  the  total  mechanical  work, 
dW,  done  on  the  body  in  any  given  short  time  will  be 

-pffdadn, 


32 


ELEMENTARY   THERMODYNAMICS. 


the  integration  extending  over  all  parts  of  the  surface 
exposed  to  the  air. 

But  when  the  only  parts  of  the  surface  at  liberty  to 
expand  are  those  which  are  exposed  to  the  normal  pressure, 
Jfdadn  will  be  the  increase  in  volume,  which  we  denote 

by  dv :  hence 

dW=-pdv (0), 

and  therefore     dU=dW+dQ  =  dQ- pdv. 

The  expression  pdv  will  be  a  complete  differential  if  p 
be  a  function  of  v  only,  or  a  constant,  or  if  v  be  constant. 
When  p  is  constant, 

dW=-d(pv), 

and  dQ 


The  mechanical  work  done  by  the  body  during  a 
change  of  volume  may  be  represented  graphically  by 
means  of  a  diagram,  first  employed  by  Watt  for  the  steam 
engine  and  often  known  as  Watt's  Diagram  of  Energy,  or 
an  Indicator  diagram.  Two  rectangular  axes  being  taken, 

P 


MN 


the  volume  of  the  body  and  the  uniform  normal  pressure 
to  which  it  is  subjected  at  any  instant  are  represented  by 
the  abscissa  and  ordinate  of  a  point  P  in  the  plane  of  the 
axes.  If,  then,  P,  Q  are  two  consecutive  positions  of  the 


THE   CONSERVATION   OF  ENERGY.  33 

point  P,  and  PM,  QN  perpendiculars  on  the  axis  of 
volumes,  the  mechanical  work  done  by  the  body  in  the 
corresponding  small  change  of  state,  which,  as  we  have 
already  seen,  is  equal  to  pdv,  will  be  proportional  to  the 
elementary  area  PMNQ.  Hence  if  PP'  denote  a  finite 
change  of  state,  the  corresponding  mechanical  work  done 
by  the  body  will  be  proportional  to  the  area  of  the  figure 
PMM'P',  and  will  evidently  depend  on  the  form  of  the 
curve  joining  PP'  as  well  as  on  the  positions  of  P  and  P' 
themselves.  If,  however,  the  normal  pressure  be  constant, 
PP'  will  be  a  straight  line  and  the  area  PMM'P  will 
depend  only  on  the  positions  of  P  and  P'.  If  the  volume 
be  constant,  the  area  PMM'P'  will  always  be  zero. 

17.  We  are  now  able  to  explain  a  practical  method 
of  measuring  non-mechanical  work,  depending  on  the  fact 
that  when  water  at  0°  C.  is  subjected  to  a  constant  normal 
pressure  of  one  atmo,  it  is  able  to  take  up  a  state  of 
mechanical  rest  either  in  the  solid  or  in  the  liquid  form. 

In  the  first  place,  it  appears  from  observation  and  may 
also  be  proved  by  calculation,  that  the  force  of  gravitation 
between  small  bodies  is  exceedingly  minute,  and  therefore 
the  work  which  it  does  will  be  insignificant — since  it  de- 
pends only  on  the  mechanical,  or  visible,  motions.  Hence 
if  we  have  a  number  of  small  bodies  near  the  surface  of 
the  earth,  whose  centres  of  mass  move  neither  up  nor 
down,  we  need  neither  consider  the  attraction  of  the 
earth  nor  their  attraction  on  one  another.  Also  by 
wrapping  the  bodies  well  up  in  the  best  non-conducting 
materials,  we  may  approximate  very  closely  to  an  ideal 
case  in  which  there  is  neither  conduction  nor  radiation  of 
heat  from  the  bodies;  that  is,  no  non-mechanical  work. 
P.  3 


34  ELEMENTARY   THERMODYNAMICS. 

Consequently,  if  these  bodies  undergo  an  operation, 
the  only  forces  with  which  we  have  to  deal  will  be  con- 
tact-forces which  do  no  non-mechanical  work.  Suppose, 
then,  that  we  take  such  a  system  consisting  of  a  quantity 
of  ice  and  two  bodies  A,  B,  exposed  only  to  the  following 
contact-forces : — 

(1)  Controllable  contact-forces  acting  on  A  and  B. 

(2)  A  constant  normal  pressure  of  one  atmo  over 
all  parts  of  the  surface  of  the  ice  which  are  at  liberty 
to  expand. 

The  system  being  originally  in  a  state  of  mechanical 
rest  at  0°  C.,  let  a  measured  quantity  of  mechanical  work, 
W  ergs,  be  expended  in  rubbing  the  two  bodies  A,  B 
together,  and  after  allowing  a  sufficient  time  for  the 
system  to  come  to  a  state  of  mechanical  rest  at  the  same 
temperature  as  before,  suppose  the  only  effect  produced 
to  be  the  conversion  of  n  grammes  of  ice  into  water. 

W 
Then  it  is  clear  that  —  is  the  quantity  of  non-mechanical 

work,  in  ergs,  that  must  be  done  on  one  gramme  of  ice  at 
0°  C.  to  convert  it  into  water  at  the  same  temperature 

W 

under  a  constant  pressure  of  one  atmo.     The  value  of  — 

n 

is  found  to  be  3,292,025,964 ;  and  since,  at  a  pressure  of 
one  atmo,  the  volume  of  one  gramme  of  ice  at  0°  C.  is 
1'087  cubic  centimetres  and  the  volume  of  one  gramme  of 
water  only  TOOOH  cubic  centimetres,  the  work  done  by 
the  pressure  of  the  air  on  the  ice  will  be  negative  and  nu- 
merically equal  to  '08689  x  1013510,  that  is,  88060,  ergs, 
per  gramme  of  ice  melted. 

Secondly,  the  non-mechanical  work  done  on  any  system 


THE  CONSERVATION  OF  ENERGY.          35 

in  any  change  of  state,  being  the  excess  of  the  total  in- 
crease of  energy  over  the  mechanical  work  done  on  the 
system,  can  be  found  as  soon  as  the  increase  of  energy  is 
known,  and  this  may  be  determined  for  conveniently  small 
bodies  in  the  following  manner : — 

(1)  In  the  initial  state,  let  the  body  be  protected 
from  all  external  influences  until  the  whole  of  the  me- 
chanical kinetic  energy  is  expended  in  internal  friction 
or  collisions.     Then  place  the  body  in  a  vessel  surrounded 
by  ice   at   0°  C.,  protected  by  the  best   non-conducting 
materials  and  subject  to  a  constant  normal  pressure  of 
one  atmo,  and  suppose  that  when  an  invariable  state  has 
been  attained  at  0°  C.,  the  only  effect  on  the  vessel  and 
the  ice  which  surrounds  it  is  that  m^  grammes  of  ice  have 
been   converted   into   water   at   the   same    temperature. 
Then,  since  the  radiant   energy  in  the  interior  of  the 
vessel  is  far  too  small  to  be  taken  into  account,  and  since 
it  is  evident  that  there  is  no  radiation  from  the  exterior 
surface  of  the  ice  and  that  the  normal  pressure  does  no 
non-mechanical  work,  the  energy  lost  from  the  given  body 
during  the  cooling  process  will  be 

ml  (3,292,025,964  -  88060)  ergs. 

(2)  In  the  final  state,  let  the  body  undergo  the 
same  processes  as  before,  and  suppose  its  ultimate  state  is 
the  same  as  its  ultimate  state  in  (1).   Then  if  ma  grammes 
of  ice  be  converted  into  water  during  the  processes,  it  is 
clear  that  the  energy  in  the  final  state  exceeds  the  energy 
in  the  initial  state  by 

(m2  -  TO!)  (3,292,025,964  -  88060)  ergs. 

18.     The  increase  of  the  non-mechanical  kinetic  energy 

3—2 


36  ELEMENTARY  THERMODYNAMICS. 

depends  only  on  the  initial  and  final  states,  while  the  non- 
mechanical  work  generally  depends  on  the  way  in  which 
the  change  of  state  takes  place,  as  well.  The  non-me- 
chanical work  done  on  a  system  is  therefore  not  generally 
equal  to  the  corresponding  increase  of  the  non-mechanical 
kinetic  energy.  This  also  follows  from  the  fact  that  the 
non-mechanical  kinetic  energy  of  a  system  may  be  altered 
by  friction  or  collision  between  the  different  parts  of  the 
system,  even  when  there  are  no  external  forces.  We  are 
consequently  unable  to  apply  the  word  '  heat '  indifferently 
to  non-mechanical  kinetic  energy  and  to  non-mechanical 
work.  Now  we  have  a  practical  method  of  determining 
non-mechanical  work ;  but  we  are  unable  to  measure 
non-mechanical  kinetic  energy,  because  we  have  never  yet 
been  able  to  deprive  a  body  of  all  its  non-mechanical 
motions.  We  shall  therefore  always  use  the  word  'heat' 
as  an  equivalent  for  non-mechanical  work,  whenever  we 
wish  to  speak  with  scientific  accuracy. 

19.  It  is  considered  to  be  proved,  by  observation  and 
experiment,  that  grittiness,  the  cause  of  friction,  is  a 
universal  property  of  matter,  and  it  may  be  concluded, 
from  the  preceding  and  similar  arguments,  that  it  is  the 
sole  cause  of  the  existence  both  of  non-mechanical  work 
and  of  non-mechanical  kinetic  energy,  and  that  it  con- 
siderably modifies  radiation;  while  the  rigid  accuracy  with 
which  all  known  actions  at  a  distance  fulfil  their  fixed 
laws  is  supposed  to  prove  that  friction  is  entirely  absent 
in  the  ether.  The  general  tendency  we  have  already 
noticed,  for  bodies  to  assume  an  apparently  invariable 
form  ^  and  internal  condition,  is  evidently  a  consequence 
of  friction.  It  will  be  seen  hereafter  to  be  a  case  of 


THE   CONSERVATION   OF   ENERGY.  37 

Carnot's  principle,  which  is  merely  a  great  Law  of  Fric- 
tion. 

20.  When  the  form  and  internal  condition  of  a  body 
have  been  rendered  constant  by  friction,  the  mechanical 
motions  of  the  body  admit  of  a  very  simple  representation. 

Let  0  be  a  point  fixed  in  the  body,  and  imagine  an 
ideal  sphere  of  unit  radius  with  0  for  centre  to  be  carried 
about  with  0  in  such  a  manner  that,  at  every  instant, 
every  point  of  its  surface  is  moving  with  a  velocity  equal 
and  parallel  to  that  of  0.  Then  as  0  moves  about,  the 
line  joining  0  to  any  point  fixed  in  the  ideal  sphere 
always  remains  parallel  to  itself.  Let  the  lines  joining  0 
to  two  points  P,  Q,  fixed  in  the  body,  meet  the  sphere,  at 
any  instant  T,  in  two  non-coincident  points  A,  B ;  and  at 
the  time  T  +  t,  in  A',  B1.  If  the  great  circle  which  bisects 
A  A'  meet  the  great  circle  which  bisects  BB'  in  /,  we  shall 
have  IA  =  IA'  and  IB  =  IB'.  Hence,  since  AB=A'R, 


the  two  triangles  TAB,  IA'B'  have  their  sides  respectively 
equal  to  one  another,  and  therefore  one  is  merely  the  same 

445246 


38  ELEMENTARY   THERMODYNAMICS. 

triangle  as  the  other  twisted  round  01.  Since  the  whole 
body  may  be  supposed  to  be  rigidly  connected  to  the 
three  points  (0,  P,  Q),  it  is  evident  that  the  mechanical 
displacement  which  has  actually  taken  place  might  have 
been  effected  in  either  of  the  following  simple  ways  :— 

(1)  First,  let  every  part  of  the  body  receive  a  dis- 
placement equal  and  parallel  to  that  of  the  point  0,  which 
will  leave  the  directions  of  OP  and  OQ  unaltered,  and 
then  rotate  the  whole  body  about  an  axis  01,  passing 
through   0,  until  the  directions  of  OP  and  OQ  change 
from  OA,  OB  to  OA',  OB',  respectively. 

(2)  Let  the  body  be  rotated  about  an  axis  passing 
through  0  in  a  direction  parallel  to  01,  through  the  same 
angle  in  the  same  direction  as  before,  and  then  give  to 
every  part  of  the  body  a  displacement  equal  and  parallel 
to  that  of  0. 

In  order  to  find  the  relation  between  the  rotations 
corresponding  to  any  two  different  base  points  0,  0',  let 
us  suppose  the  point  0  to  carry  about  with  it  three 
rectangular  axes  whose  directions  always  remain  parallel 
to  themselves,  Oz  being  the  axis  of  rotation  through  0 
and  the  plane  zOy  passing  through  the  position  of  the 
point  0'  at  the  time  T.  If  the  figure  represent  the  posi- 
tions of  these  axes  at  the  time  T+  t,  and  if  Q1}  Q2  be  the 
positions  of  0'  relatively  to  the  axes  at  the  times  T,  T  +  t, 
the  points  Qlt  Q2  will  lie  on  a  circle  whose  centre  is  M,  the 
point  where  the  plane  through  Q^  and  Q2  parallel  to  xOy 
meets  Oz.  If  Q  be  the  original  position  of  0',  then  when 
0  is  chosen  for  base  point,  every  part  of  the  body  receives 
a  displacement  equal  and  parallel  to  QQl}  in  consequence 
of  which  Oz  is  brought  into  the  position  shown  in  the 


THE   CONSERVATION    OF   ENERGY. 


39 


figure,  and  afterwards  the  whole  body  is  rotated  about  Oz 
through  an  angle  6  =  QjMQ2,  When  0'  is  chosen  for  base 
point,  every  part  of  the  body  first  receives  a  displacement 


equal  and  parallel  to  QQ2,  which  is  equivalent  to  a  dis- 
placement QQl  followed  by  another  displacement  QjQ2. 
Hence,  when  0'  is  the  base  point,  the  axis  Oz  will  be 
brought  by  the  motion  of  translation  into  a  parallel  posi- 
tion passing  through  N,  where  Q^MN  is  a  parallelogram. 
To  bring  Oz  into  its  final  position,  we  must  then  rotate 
the  body  in  the  same  direction  and  through  the  same 
angle  as  before  about  an  axis  parallel  to  Oz,  passing 
through  Q2,  the  final  position  of  0'.  Thus  the  axes  of 
rotation  corresponding  to  all  base  points  are  parallel  and 
the  angles  of  rotation  equal  and  in  the  same  direction. 
Hence,  at  any  instant,  the  velocity  of  any  part,  P,  of 


40  ELEMENTARY   THERMODYNAMICS. 

the  body  is  the  resultant  of  the  velocity  of  any  base  point 
0,  fixed  in  the  body,  and  the  velocity  that  P  would  have 
if  0  were  at  rest  and  the  body  rotating  about  an  axis  01, 
passing  through  0.  If  a  different  base  point,  0',  be  chosen, 
the  axis  of  rotation  through  0'  will  be  parallel  to  01  and 
the  angular  velocity  about  it  equal  to  that  about  01  and 
in  the  same  direction. 

For  the  further  discussion  of  rotations,  the  principle  of 
Angular  Momentum  is  required. 

21.  Let  Ox,  Oy,  Oz  be  any  three  rectangular  axes,  and 
let  P  be  the  position  of  a  particle  whose  mass  is  m. 
Suppose  the  plane  through  P  parallel  to  xOy  to  meet 
Oz  in  N,  arid  let  PFbe  the  direction  of  the  velocity  of 


the  particle  in  this  plane.  Then  we  define  the  Angular 
Momentum,  or  the  Moment  of  the  Momentum,  of  the 
particle  at  P  about  Oz  to  be  the  product  of  the  resolved 
linear  momentum  along  PF  and  the  perpendicular  NK 
from  the  point  N  on  PF,  the  product  being  reckoned 
positive  when  the  particle's  velocity  tends  to  carry  it 
round  Oz  in  the  positive  direction  and  vice  versa.  The 
positive  directions  round  the  axes  are  always  taken  to  be- 
round  Ox,  from  y  to  z;  round  Oy,  from  z  to  as;  and  round 
Oz,  from  x  to  y. 


THE  CONSERVATION  OF  ENERGY.         41 

The  angular  momentum  round  Oz  of  a  finite  body,  or 
of  a  system  of  bodies,  is  defined  to  be  the  sum  of  the 
angular  momenta  of  its  ultimate  particles  about  Oz. 
When  a  system  contains  several  bodies,  its  angular 
momentum  about  any  line  will  therefore  be  equal  to 
the  sum  of  the  angular  momenta  of  the  separate  bodies. 

The  moment  of  a  force  about  any  line  Oz  is  defined  in 
the  same  way  as  the  angular  momentum  of  a  particle, 
linear  momentum  being  simply  replaced  by  force. 

If  e  be  the  angle  between  NK  and  NP,  r  the  length 
of  NP,  v  the  velocity  of  the  particle  along  P  V,  and  m  its 
mass,  its  angular  momentum  about  Oz  will  be  mv  (r  cos  e). 


Hence,  since  m  (v  cos  e)  is  the  resolved  linear  momentum 
perpendicular  both  to  Oz  and  NP,  we  see  that  the  angular 
momentum  of  the  particle  about  Oz  is  equal  to  the  product 
of  the  perpendicular  distance,  PN,  of  the  particle  from 
Oz,  into  the  resolved  linear  momentum  at  right  angles 
both  to  Oz  and  NP,  reckoned  positive  when  in  the  positive 
direction  round  Oz. 

Again,  if  P  V  represent  the  magnitude  of  the  resolved 
linear  momentum  as  well  as  its  direction,  it  is  clear  that 
the  angular  momentum  of  the  particle  about  Oz  will  be 


42  ELEMENTARY   THERMODYNAMICS. 

proportional  to  the  area  of  the  triangle  NPV.  Now  if 
Pa,  Pb,  PC,  be  the  components  of  the  total  linear  mo- 
mentum of  P  in  any  three  directions,  according  to  the 


parallelogramic  law,  the  sum  of  the  perpendiculars  from 
(a,  b,  c)  on  a  plane  through  NP  and  Oz  will  be  equal  to 
the  perpendicular  from  V  on  NP.  Hence  if  (a',  b',  c')  be 
the  projections  of  (a,  b,  c)  on  the  plane  NPV,  the  sum  of 
the  perpendiculars  from  (a',  b',  c')  on  NP  will  be  equal  to 
that  from  V.  Consequently,  the  triangle  NPV  is  equal 
to  the  sum  of  the  three  triangles  NPa',  NPb',  NPc' ;  or 
the  angular  momentum  of  a  particle  about  any  line  Oz  is 
equal  to  the  sum  of  the  moments  of  its  component  linear 
momenta.  Similarly  it  may  be  shown  that  the  moment 
of  a  force  is  equal  to  the  sum  of  the  moments  of  its 
components. 

Next,  let  P,  P'  be  the  positions  of  the  particle  at  two 
different  times  t,  t  +  r,  where  r  is  indefinitely  small.  Let 
PV  represent  the  total  linear  momentum  at  the  time  if, 
and  suppose  the  particle  to  be  acted  on  at  that  instant  by 
a  resultant  force  F  in  the  direction  PA.  Also  let  the 
three  components  of  the  total  linear  momentum  at  the 


THE  CONSERVATION  OF  ENERGY.          43 

time  t  +  T,  according  to  the  parallelogramic  law,  be  repre- 
sented by : — 

(1)  P'V,  equal  and  parallel  to  PV. 

(2)  PA' ,  representing  FT,  in  a  direction  parallel  to 
PA. 

(3)  P'B',  the   magnitude   and   direction  of  which 
depend  on  the  way  in  which  the  force  which  acts  on  the 
particle  varies  as  the  particle  moves  from  P  to  P. 


V7 


Then  if  q,  q   be  the  angular  momenta  of  the   particle 

about  any  line  Oz  at  the  times  t,  t  +  T,  we  have 

q  =  moment  of  P'  V  +  moment  of  P'A'  +  moment  ofP'R, 

q  =  moment  of  PV. 

But  P'B'  and  the  distance  between  PV  and  P'V  are 

quantities  of  the  second  order:  hence  if  wre  retain  only 

quantities  of  the  first  order,  we  get 

q  —  q  =  moment  of  P'A'. 

Now  the  moment  of  P'A  may  be  taken  to  be  the  same 
as  the  moment  of  FT  acting  at  P  in  the  direction  PA, 


44  ELEMENTARY   THERMODYNAMICS. 

because  their  difference  is  a  small  quantity  of  the  second 
order :  hence  ^—3  —  moment  about  Oz  of  the  force  F 

T 

acting  at  P.  Proceeding  to  the  limit,  we  see  that,  at  any 
instant,  the  rate  at  which  the  angular  momentum  of  the 
particle  about  any  line  is  increasing  with  the  time  is  equal 
to  the  moment  of  the  resultant  force  which  acts  upon  it  at 
that  instant.  It  follows  therefore  that  the  rate  at  which 
the  angular  momentum  about  any  straight  line  of  a  finite 
body,  or  of  a  system  of  bodies,  increases  with  the  time,  is 
equal  to  the  sum  of  the  moments  about  this  straight  line 
of  all  the  forces,  both  external  and  internal,  which  act  upon 
it.  But,  according  to  the  third  law  of  motion,  the  internal 
forces  consist  of  a  set  of  equal  and  opposite  reactions,  and 
consequently  the  sum  of  their  moments  is  zero.  Hence 
the  rate  at  which  the  angular  momentum  increases  with 
the  time  is  simply  equal  to  the  moment  of  the  external 
forces.  If,  then,  the  moment  of  the  external  forces  about 
any  straight  line  is  constantly  zero,  the  angular  mo- 
mentum of  the  system  about  this  straight  line  will  remain 
constant. 

22.  Suppose  the  annexed  figure  to  represent  the 
projections  of  the  various  lines  on  a  plane  through  G,  the 
centre  of  mass,  parallel  to  xOy.  Let  QR  be  the  projection 
of  the  direction  of  the  velocity  of  a  particle  P  whose  mass 
is  m.  Draw  OQ  and  GR  perpendicular  to  QR,  and  let  a 
line  through  G  parallel  to  RQ  meet  OQ  in  S.  Then  if  v 
be  the  velocity  of  the  particle  along  QR,  its  angular  mo- 
mentum about  Oz  will  be  mv.  OQ,  or  mv .  GR  +  mv .  OS. 
The  term  mv.GR  is  the  angular  momentum  of  the 
particle  about  an  axis  through  G  parallel  to  Oz :  the  other 


THE  CONSERVATION  OF  ENERGY. 


45 


term,  mv .  OS,  is  the  angular  momentum  about  Oz  of  a 
particle  of  mass  m  situated  at  G  and  moving  with  a 
velocity  equal  and  parallel  to  that  of  P.  The  angular 

o 


momentum  of  the  whole  system  about  Oz  is  therefore 
equal  to  the  angular  momentum  about  a  parallel  axis 
through  G,  together  with  what  the  angular  momentum 
about  Oz  would  be  if  all  the  particles  were  transferred  to 
G  without  altering  their  velocities  either  in  magnitude  or 
direction.  But,  since  G  is  the  centre  of  mass,  these  ideal 
particles  at  G  are  known  to  be  equivalent,  as  to  linear 
momentum,  to  a  single  particle  of  mass  M  situated  at  G 
and  moving  about  with  it,  M  being  the  mass  of  the  whole 
system.  Hence,  since  the  angular  momentum  of  a  particle 
is  equal  to  the  sum  of  the  moments  of  its  component 
linear  momenta,  it  follows  that  the  angular  momentum  of 
the  system  about  Oz  is  equal  to  the  angular  momentum 
about  a  parallel  axis  through  G,  together  with  the  angular 
momentum  about  Oz  of  a  mass  M  placed  at  G  and  moving 
about  with  it.  We  shall  refer  to  these  two  parts  as  the 
angular  momenta  due  to  rotation  and  translation,  respec- 
tively. 

In  like  manner  it  may  be  shown  that  the  moment  of  a 


46  ELEMENTARY   THERMODYNAMICS. 

force  about  Oz  is  equal  to  its  moment  about  a  parallel  axis 
through  G  (or  any  other  point  G'\  together  with  the 
moment  about  Oz  of  an  equal  and  parallel  force  acting  at 
G  (or  G').  Thus  the  moment  of  all  the  external  forces 
about  Oz  is  equal  to  their  moment  about  a  parallel  axis 
through  G  (or  G'},  together  with  the  moment  about  Oz  of 
the  resultant  of  a  system  of  equal  and  parallel  forces  acting 
at  G  (or  G'). 

As  the  body  moves  about,  let  G,  G',  be  the  positions  in 
space  of  the  centre  of  mass  at  any  two  consecutive  instants 
t,  t  +  r ;  and  let  GN,  G'N'  be  the  corresponding  positions 
of  the  straight  line  drawn  through  G  parallel  to  the  fixed 
line  Oz.  Also  let  q  be  the  angular  momentum  of  the  body 
about  GN  at  the  time  t,  and  q'  the  angular  momentum 
about  G'N'  at  the  time  t  +  r.  Then  since  the  angular 
momentum  of  the  body  about  GN  at  the  time  t  +  T  is 
equal  to  the  sum  of  q  and  of  the  moment  of  a  mass  M 
situated  at  G  and  moving  about  with  it,  it  is  evident  that 
the  angular  momentum  about  GN  at  the  time  t  +  T  differs 
from  q  only  by  a  small  quantity  of  the  second  order,  and 
that  the  rate  at  which  the  angular  momentum  about  a 
fixed  straight  line,  coinciding  with  GN,  is  increasing  with 

the  time  at  the  time  t,  is  equal  to  the  limit  of  ^-— ^ .     We 

therefore  see  that  at  any  instant,  the  moment  of  the 
external  forces  about  the  moving  axis  through  G  parallel 
to  Oz  is  equal  to  the  rate  at  which  the  angular  momentum 
about  that  axis  is  then  increasing  with  the  time,  that  is,  is 
equal  to  the  rate  of  increase  of  the  angular  momentum 
of  rotation.  From  this  it  follows  that  the  moment  about 
Oz  of  a  system  of  forces  applied  at  G,  equal  and  parallel  to 
the  external  forces,  must  be  equal  to  the  rate  of  increase 


THE   CONSERVATION   OF   ENERGY. 


of  the  angular  momentum  of  translation.  Hence,  when 
there  are  no  external  forces,  the  angular  momenta,  both 
of  translation  and  rotation,  remain  constant. 

Again,  the  velocity  of  any  particle  P  may  be  supposed 
to  consist  of  a  velocity  equal  and  parallel  to  that  of  G,  the 
centre  of  mass,  combined  with  the  velocity  of  P  relative 
to  G.  If  p  be  the  distance  of  P  from  a  plane  through  the 


direction  of  G's  velocity  perpendicular  to  xOy,  the  angular 
momentum  due  to  the  former  component  about  an  axis 
through  G  parallel  to  Oz  will  be  mpV,  where  V  is  the 
velocity  of  G  parallel  to  the  plane  xOy.  For  the  whole 
system,  this  is  V2,mp,  distances  on  one  side  of  the  plane 
through  G  being  considered  positive,  on  the  other  negative. 
But  by  a  well  known  property  of  the  centre  of  mass, 
Smp  =  0.  Hence,  finally,  the  angular  momentum  of  ro- 
tation depends  only  on  the  velocities  of  the  particles 
relative  to  the  centre  of  mass. 

In  consequence  of  these  important  properties,  we  shall 
always  suppose  the  mechanical  motions  of  a  body  of  which 
the  form  and  internal  condition  are  sensibly  invariable,  to 
consist  of  the  motion  of  translation  of  the  centre  of  mass 
combined  with  a  rotation  about  an  axis  passing  through 
the  centre  of  mass. 


48 


ELEMENTARY   THERMODYNAMICS. 


23.  The  preceding  principles  immediately  lead  to 
some  very  valuable  results  relating  to  bodies  of  a  perma- 
nent internal  condition.  Thus  let  a  body  free  from 
mechanical  vibrations  be  rotating  with  angular  velocity 
a  about  an  axis  passing  through  G,  the  centre  of  mass. 
Take  three  rectangular  axes  Gx,  Gy,  Gz,  one  of  which, 
Gz,  coincides  with  the  axis  of  rotation.  Then  any  small 
portion  of  the  body  will  describe  a  circle  about  G  in  a 
plane  parallel  to  xGy.  Hence  if  r  be  the  distance  of  any 


particle  P  from  Gz,  its  mechanical  motion  relative  to  G 
will  simply  consist  of  a  velocity  cor  at  right  angles  both 
to  Gz  and  to  the  perpendicular  from  P  on  Gz.  Besides 
this,  there  may  be  velocities  (a,  /8, 7)  relative  to  G  parallel 
to  the  axes,  due  to  the  non-mechanical  motions.  The 
angular  momentum  about  Gz  of  the  motion  of  P  relative 
to  G,  being  the  sum  of  the  moments  of  the  component 
linear  momenta  relative  to  G,  will  therefore  be 

mr-o)  +  m  (fix  —  ay), 
where  m  is  the  mass  of  the  particle. 


THE  CONSERVATION  OF  ENERGY.          49 

The  angular  momentum  of  the  whole  body  about  Gz  is 
consequently  equal  to 

toSwr2  +  2m  (/8a?  —  ay). 

But  the  angular  momentum  is  clearly  zero  when  the 
body  does  not  rotate,  that  is,  when  to  =  0,  and  the  non- 
mechanical  motions  are  the  same  as  before.  Hence 

2m  (fix  -  ay)  =  0, 

and  if  C  be  the  '  Moment  of  Inertia '  of  the  body  about 
Gz,  or  (7=2mr2,  the  angular  momentum  about  the  axis 
of  rotation  takes  the  very  simple  form  Cw. 

It  is  usual  to  put  C  =  Mk*,  where  M  is  the  mass  of 
the  whole  body.  The  length  k  is  known  as  the  '  radius  of 
gyration'  about  Gz,  and  it  is  evident  that  the  angular 
momentum  of  the  body  about  Gz  is  the  same  as  that  of 
an  ideal  particle  of  mass  M  situated  at  a  distance  k 
from  Gz  and  revolving  round  Gz  with  angular  velocity  &>. 

The  angular  momenta  of  the  body  about  Gx  and  Gy 
are  not  necessarily  zero,  although  the  body  is  rotating 
about  an  axis  at  right  angles  to  both  of  them.  In  fact, 
since  the  velocity  wr  is  equivalent  to  —  coy  parallel  to  Gx 
and  +  wx  parallel  to  Gy,  the  angular  momentum  about 
Gx  is  equal  to 

—  a&mxz  -f  2w  (jy  —  /3z). 

The  second  term  may  be  shown  to  be  zero  as  before,  and 
thus  the  final  result  is  —  wZmxz.  Similarly  the  angular 
momentum  about  Gy  is  —  a&myz. 

If  the  centre  of  mass  be  in  motion,  as  in  the  case  of  a 

planet  revolving  round  the  sun,  and  if  fi  be  its  angular 

velocity  about  an  axis  Oz    parallel  to  Gz,  the  angular 

momentum  of  translation  about  Oz'  will  be  MR2fl,  where 

P.  4 


50  ELEMENTARY  THERMODYNAMICS. 

R  is  the  distance  of  G  from    Oz'.     The   total    angular 
momentum  about  Oz'  is  therefore 

To  find  the  kinetic  energy,  let  (u',  v,  w'}  be  the  veloci- 
ties of  P,  (u,  v,  w)  those  of  G,  so  that 

v  =  v  +  wx  +  /8  >. 
w'  =  w  +  7         J 
Then 

1 2m  (u-  +  v'2  +  w/2)  =  -|  (w2  +  w2  +  w2)  2m 

(rf 


2m  (/&e  — 
But  we  have 

2w  (ySa;  -  ay}  =  0, 

2m#  =  2my  =  0,  since  (r  is  the  origin  ; 
also          2mtt'  =  2mw,  or  2ma  =  0, 
2mi/  =  2mv,  or  2m/S  =  0, 
2m^'  =  2mw,  or  2m-7  =  0, 
and  2m  =  J/. 

Hence  we  obtain 


(11). 

The  term  %M  (u?  +  v*  +  w*)  is  called  the  mechanical 
kinetic  energy  of  translation,  being  the  same  as  the 
kinetic  energy  of  a  single  particle  of  mass  M  moving 
about  with  Q,  the  centre  of  mass.  The  term  %Ca>\ 
or  IJlf^eo2,  is  called  the  mechanical  kinetic  energy  of 
rotation,  and  is  the  same  as  the  kinetic  energy  of  a  single 
particle  of  mass  M  moving  with  velocity  kca. 


THE   CONSERVATION    OF   ENERGY.  51 

24.  Again,  if  at  any  instant  t,  a  body  be  rotating 
about  a  straight  line  Gz,  those  parts  of  the  body  which 
then  lie  on  Gz  will  all  be  moving  with  velocities  equal  and 
parallel  to  that  of  G.  Hence  in  any  short  interval  r,  the 
displacements  of  these  parts  will  be  equal  and  parallel  to 
that  of  G,  and  thus,  at  the  instant  t  +  r,  they  will  still  lie 
on  a  line  through  G  parallel  to  the  original  direction  of  Gz. 
If,  therefore,  at  the  time  t  +  r,  the  body  be  rotating  about 
an  axis  Gz',  drawn  through  G  in  a  different  direction  from 
Gz,  it  is  clear  that  Gz  and  Gz'  cannot  have  the  same 
position  with  respect  to  the  body.  This  fact  is  usually 
expressed  by  saying  that  when  the  axis  of  rotation  moves 
about  in  space,  it  also  moves  about  in  the  body.  In  like 
manner  it  may  be  shown  that  when  the  axis  of  rotation 
is  fixed,  as  to  direction,  in  space,  it  is  also  fixed  in  the 
body. 

Now  it  is  evident  that  the  internal  mechanical  stresses 
of  a  body  depend  only  on  its  form  and  internal  condition, 
and  therefore  when  these  are  invariable,  the  stresses  will 
be  constant  in  magnitude  and  in  directions  which  are 
fixed  with  regard  to  the  body.  A  little  consideration 
tells  us  that,  when  there  are  no  external  forces,  this 
condition  is  inconsistent  with  a  change  of  the  axis  of 
rotation.  Thus  a  body  which  is  under  the  action  of  no 
forces  and  possesses  angular  momentum  about  any  straight 
line  through  G,  must  either  be  rotating  about  an  axis 
through  G  fixed  in  the  body  and  in  a  constant  direction 
in  space  or  be  in  a  state  of  mechanical  vibration.  Now 
if  we  assume  the  tendency  of  bodies  under  the  action  of 
no  force  to  take  an  invariable  internal  condition  to  be 
universal,  it  follows  that  the  mechanical  vibrations  will 
all  ultimately  disappear.  But  since  there  are  no  external 

4—2 


r>2 


ELEMENTARY   THERMODYNAMICS. 


forces,  the  angular  momentum  of  the  body  about  any 
straight  line  drawn  in  a  fixed  direction  through  G,  the 
centre  of  mass,  is  constant.  The  mechanical  motions  of 
the  body  relative  to  G  cannot  therefore  all  disappear  and 
we  conclude  that  the  body  will  ultimately  settle  down 
into  a  state  of  steady  rotation  about  an  axis  fixed  in  the 
body  which  passes  through  G  in  a  constant  direction. 

When  a  body  subject  to  no  external  influences  has 
taken  up  its  final  state  of  rotation  about  an  axis  Gz,  let 
the  point  G  be  imagined  to  carry  about  with  it  three 
rectangular  axes  Ga,  Gy,  Gz,  whose  directions  always 
remain  parallel  to  themselves.  Then  since  Ceo  is  the 
angular  momentum  about  Gz,  &>  will  be  constant,  and 
since  —  a&mxz  is  the  angular  momentum  about  Gx,  a 
line  drawn  in  a  fixed  direction,  ^mxz  will  also  be  constant. 
Now  if  Ox',  Gy'  be  any  two  axes  at  right  angles  to  one 
another  and  to  Gz  which  are  carried  round  with  the  body, 


Smas'z  and  tmy'z  will  evidently  be  constant.     Also  if  0 
be  the  angle  between  Gx  and  Oaf,  we  shall  have 
x  =  x  cos  6  +  y'  sin  6, 


and  therefore 


sin  O^my'z. 


THE   CONSERVATION    OF   ENERGY.  53 

Since  6  will  pass  through  every  value  as  the  axes  Gx ',  Gy 
revolve  round  Gz,  this  result  shows  that  Gz  makes 

'Zmzx  =  "Zmzy1  =  0, 
and  ^mzx  =  "Zmzy  =  0, 

or  the  angular  momentum  of  the  body  about  any  line  at 
right  angles  to  the  axis  of  rotation  is  zero. 

25.  We  can  now  explain  some  further  results  which 
are  of  the  highest  interest  and  importance  in  thermo- 
dynamics. 

First,  suppose  all  the  external  forces  which  act  on  a 
body  to  be  actions  at  a  distance  such  that  the  force  on 
any  particle  is  equal  to  mg,  where  at  any  instant,  g  is  the 
same  in  magnitude  and  direction  for  every  particle  of  the 
body.  Then,  since  it  is  clear  that  such  forces  have  no 
effect  in  producing  internal  stresses,  and  since  they  give 
no  moment  about  any  axis  through  G,  the  centre  of  mass, 
the  body  will  behave,  as  to  rotations,  exactly  as  if  it 
were  under  the  action  of  no  forces.  It  will  therefore 
settle  down  at  last  into  a  state  of  uniform  rotation  about 
an  axis  passing  through  G  in  a  constant  direction.  The 
mechanical  kinetic  energy  of  the  body  will  then  be 

%M  (u2  +  tf  +  w2)  +  %  Cm2, 

where  M  is  the  mass  of  the  body,  &>  the  uniform  angular 
velocity,  and  (u,  v,  w)  the  velocities  of  G  parallel  to  three 
rectangular  axes  Ox,  Oy,  Oz,  fixed  in  space. 
But  if  g  be  equivalent  to  (gx,  gu,  gt)  parallel  to  these 
axes,  we  have 

,,  du 

x  +  m3gx  +  . . .  =  M  -^  , 

du 


54  ELEMENTARY   THERMODYNAMICS. 

Again,  the  work  done  on  the  body  by  the  forces  parallel 
to  Ox.  in  the  short  time  dt,  will  be 

Now  if  (x,  y,  z)  be  the  coordinates  of  G,  we  have 

m^Xi  +  m^x2  +  m3x3  +. . .  =  MX, 
and  therefore 


Thus  the  total  work  done  on  the  body  by  the  external 
forces,  in  any  finite  time,  is 


Hence,  since  C  and  <w  are  both  constant,  and  all  the  work 
done  by  the  external  forces  in  this  case  is  mechanical 
work,  we  see  that  the  mechanical  work  done  on  the 
body  is  equal  to  the  increase  of  its  mechanical  kinetic 
energy. 

The  problem  we  have  just  considered  is  very  nearly 
that  of  small  bodies  falling  in  a  vacuum  at  the  surface  of 
the  earth  ;  where  the  magnitude  of  g  is  the  same  both  for 
all  particles  and  for  all  time,  but  its  direction,  though  at 
any  instant  the  same  for  all  particles,  is  not  independent 
of  the  time,  being  carried  about  with  the  earth.  The 
problem  is  also  that  of  the  whole  earth,  roughly  speak- 
ing, and  consequently  the  mechanical  irregularities  of  its 
motion,  as  shown  in  the  tides,  &c.,  are  insignificant  for 
so  large  a  body.  In  the  case  of  the  sun,  there  are  dis- 
turbing effects  of  enormous  importance  due  to  radiation- 
forces,  by  which  the  whole  mass  is  kept  in  a  state  of 
tumult. 


THE  CONSERVATION  OF  ENERGY.         55 

Again,  suppose  all  the  external  forces  which  act  on  a 
body  to  be  contact-forces  of  constant  magnitude  whose 
positions  with  respect  to  any  three  rectangular  axes  in 
fixed  directions  at  G  are  invariable,  the  forces  being  such 
that  they  give  no  moments  about  these  axes  and  do  no 
non-mechanical  work.  Then,  since  there  are  no  radiation- 
forces,  if  the  body  had  no  angular  momentum  before  the 
application  of  the  forces,  it  is  evident  that,  after  they  are 
applied,  it  will  ultimately  assume  an  apparently  invariable 
form  and  internal  condition  without  rotation.  When  this 
is  the  case,  the  displacement  of  every  part  of  the  surface 
will  be  equal  and  parallel  to  that  of  G.  Hence  if 
(X,  Y,  Z}  be  the  total  external  forces  parallel  to  three 
fixed  rectangular  axes  Ox,  Oy,  Oz,  the  mechanical  work 
which  they  do  on  the  body  in  any  short  time  will  be 

Xdx  +  Ydy  +  Zdz, 

where  (x,  y,  z\  (x  +  dx,  y  +  dy,  z  +  dz)  are  the  initial  and 
final  positions  of  G. 

Since  X  =  M  -,-  ,  &c.,  &c.,  this  is  equal  to 
du       dv         < 


Hence  in  any  finite  time  the  mechanical  work  done  on 
the  body  will  be 

Pf  (11?  +  vf  +  w?)  -  $M  (u*  + 1>08  +  Wo3). 

Thus  there  is  again  no  change  in  the  internal  condition  of 
the  body  and  the  mechanical  work  done  upon  it  is  exactly 
equal  to  the  increase  of  its  mechanical  kinetic  energy, 
while  the  non-mechanical  work  is  zero. 


56 


ELEMENTARY  THERMODYNAMICS. 


In  both  cases,  it  will  be  seen  that  the  body  is  subjected 
to  a  non-frictional  process  in  which  no  non-mechanical 
work  is  done  upon  it  and  that  the  only  effect  produced 
is  a  change  in  the  motion  of  the  centre  of  mass. 

In  general,  when  a  body  is  acted  on  by  external  forces, 
the  internal  condition  of  the  body  will  not  be  constant 
nor  the  increase  of  its  mechanical  kinetic  energy  equal  to 
the  mechanical  work  done  upon  it.  Suppose,  for  example, 
that  a  body  continues  to  rotate  about  an  axis  Gz,  drawn 
through  G  in  a  constant  direction,  under  the  action  of 
contact-forces  whose  moment  about  Gz  is  N  and  about 
the  other  axes  at  right  angles  to  Gz  zero.  Also  suppose 
that  the  sums  of  the  components  of  the  external  forces 
parallel  to  the  axes  are  constantly  zero  and  that  the  point 
G  is  at  rest.  Then  if  F  be  the  force  on  any  small  element 
of  the  surface  at  P  in  the  direction  of  rotation  at  right 
angles  both  to  Gz  and  PN,  the  perpendicular  from  P  on 


Gz,  the  mechanical  work  done  by  this  force  in  the  short 
time  dt  will  be  Frudt,  where  r  is  the  distance  PN  and  co 
the  angular  velocity  of  the  body.  Hence  the  mechanical 
work  done  on  the  body  by  all  the  external  forces  will 
>e  a>dt2Fr,  or  N<odt.  Now  N  is  the  rate  of  increase  of 
the  angular  momentum  about  Gz,  that  is,  N  =  -(Co>). 


THE   CONSERVATION    OF   ENERGY.  57 

Thus  the  mechanical  work  done  on  the  body  in  a  finite 
time  is 


The  increase  of  the  mechanical  kinetic  energy  of  the  body 


These  will   not   necessarily  be  equal   except  when  ca-  ~ 

is  constantly  zero,  which  will  not  generally  be  true. 

If  the  body  be  made  of  very  rigid  materials,  like  a 
fly-wheel,  the  variations  of  C  will  be  extremely  small,  and 
therefore,  if  the  friction  of  the  axle  be  neglected,  the 
mechanical  work  done  upon  the  wheel  will  be  practically 
equal  to  the  increase  of  its  mechanical  kinetic  energy. 

The  rest  of  the  chapter  will  be  devoted  to  the  experi- 
mental evidence  of  the  principle  of  energy.  And,  for  the 
future,  as  we  shall  seldom  have  any  further  occasion  to 
make  use  of  the  conceptions  of  'kinetic  energy'  and 
'work,'  in  the  strict  sense  of  the  terms,  we  shall  fre- 
quently follow  the  popular  usage  by  employing  these 
expressions  instead  of  the  more  correct,  but  somewhat 
longer  forms, '  mechanical  kinetic  energy '  and  '  mechanical 
work.'  Also  when  we  speak  about  forces,  we  shall  always 
refer  to  their  mechanical  effect :  if  we  wish  to  refer  to 
the  non-mechanical  effect,  we  shall  always  use  the  word 
'  heat.' 


58  ELEMENTARY   THERMODYNAMICS. 

26.  We  have  seen  that  the  energy  of  a  material 
system  in  any  state  P  is  equal  to  the  sum  of  the  kinetic 
energies  of  its  ultimate  particles  and  of  the  work  which 
the  internal  ether-forces  do  on  the  system  as  it  returns  in 
any  way  from  the  state  P  to  a  fixed  standard  state  0. 
Now  since  we  possess  no  instruments  by  means  of  which 
the  motions  of  the  individual  particles  can  be  examined 
and  since  gravitation,  electric,  and  magnetic  forces  are 
the  only  ether-forces  about  which  much  is  known,  it  is 
obvious  that  we  have  no  practical  method  of  obtaining  all 
the  details  which  have  been  discussed  theoretically  in  the 
previous  part  of  the  chapter.  In  fact,  for  experimental 
purposes,  the  state  of  a  material  system  is  considered  to 
depend  on  the  following  observable  particulars  only : — 

(1)  The  chemical  and  physical  states  and  relative 
positions  of  the  different  parts  of  the  system. 

(2)  Their  mechanical  motions. 

(3)  Their  temperatures. 

(4)  Their  electric  and  magnetic  states. 

On  account  of  the  importance  of  temperature,  it  is 
necessary  that  we  should  have  correct  ideas  on  the  subject 
and  that  we  should  understand  the  method  of  constructing 
the  mercurial  thermometer. 

When  any  system  of  two  bodies  A,  B  is  protected 
from  all  external  influences  until  the  ultimate  invariable 
state  is  attained,  the  two  bodies  A,  B  are  then  said  to  be 
at  the  same  temperature.  In  order  to  test  equality  of 
temperatures,  we  may  take  a  third  body,  C,  as  an  ordinary 
mercury  thermometer,  whose  changes  of  state  are  easily 
visible  to  the  eye,  and  put  it  in  contact,  first  with  A  and 
then  with  R  If  the  bodies  be  protected  from  all  ex- 


THE  CONSERVATION  OF  ENERGY.          59 

terual  influences,  as  before,  and  C  assumes  the  same  final 
state  in  both  cases,  the  temperatures  of  A  and  B  are 
equal. 

If  the  two  bodies  A,  B,  when  first  placed  in  contact, 
do  not  at  once  take  up  an  invariable  state,  their  tempe- 
ratures are  not  necessarily  different.  Thus  salt  and  snow 
may  be  at  the  same  temperature,  as  shown  by  the  ther- 
mometer, yet  when  they  are  mixed  together,  a  violent 
chemical  action  takes  place  by  which  both  are  partially 
melted. 

The  mercurial  thermometer  is  the  instrument  most 
commonly  used  for  detecting  differences  of  temperature. 
It  consists  of  a  closed  glass  vessel  containing  only  mercury 
and  the  vapour  of  mercury.  The  glass  vessel  is  in  the 
form  of  a  fine  tube,  or  stem,  terminating  in  a  much  wider 
part,  called  the  bulb,  and  the  whole  of  the  bulb  and  part 
of  the  stem  is  filled  with  mercury  in  the  liquid  state.  On 
heating  the  thermometer,  the  glass  expands,  but  the 
mercury  expands  more,  so  that  it  is  forced  to  rise  in  the 
tube,  thereby  giving  a  visible  effect  of  change  of  tempe- 
rature. 

The  construction  of  an  accurate  thermometer  requires 
five  processes,  which  may  be  explained  as  follows. 

(1)  In  the  first  place  we  procure  a  tube  of  as  uniform 
bore  as  possible,  and  then  calibrate  it,  that  is,  divide  it 
into  short  lengths  of  equal  capacity.  To  do  this,  we  force 
a  short  column  of  mercury  into  the  tube  and  mark  the 
tube  at  both  ends  of  the  column,  then  move  it  on  its  own 
length  until  one  end  comes  exactly  to  where  the  other 
was  and  mark  the  other  end,  and  so  on.  After  this 
operation,  the  bulb  A  is  blown  at  one  end  of  the  tube  to 
a  little  more  than  the  required  size  and  closed,  and,  to 


60  ELEMENTARY   THERMODYNAMICS. 

enable  us  to  introduce  the  mercury,  a  temporary  bulb  B 
is  blown  at  the  other  end  and  drawn  out  to  a  point  at 
which  there  is  a  small  hole.  In  forming  the  bulbs,  the 
air  must  not  be  blown  by  the  mouth,  for  fear  of  the  effects 
of  moisture. 

(2)  Both  bulbs  are  now  gently  heated,  in  order  to 
expel  some  of  the  air  with  which  they  are  filled.  The 
point  of  B  is  then  immersed  in  a  vessel  containing 
mercury.  As  the  air  within  the  bulbs  cools,  its  expansive 


force  diminishes,  so  that  a  quantity  of  mercury  is  forced 
into  the  bulb  B  by  the  pressure  of  the  atmosphere.  The 
instrument  is  then  set  upright,  and  by  alternately  heating 
and  cooling  the  permanent  bulb  A,  a  sufficient  quantity 
of  mercury  is  caused  to  descend  into  it  through  the  narrow 
tube  from  the  bulb  B. 

(3)  The  instrument  is  next  laid  in  a  sloping  position, 
with  the  bulb  A  lowest,  on  a  special  furnace  and  heated 
till  the  mercury  boils.  The  vapour  of  the  boiling  mercury 
drives  out  the  air,  so  then  when  the  boiling  ceases,  the 
mercury  runs  back  from  the  bulb  B  and  completely  fills 


THE   CONSERVATION   OF    ENERGY.  61 

the  tube  as  well  as  the  bulb  A.  Then,  while  the  thermo- 
meter is  still  hotter  than  any  temperature  at  which  it  is 
afterwards  to  be  used  and  the  whole  of  the  tube  filled 
with  mercury,  the  top  of  the  tube  is  melted  and  closed  by 
a  blowpipe  flame.  As  the  thermometer  cools,  the  mercury 
contracts  and  leaves  a  space  at  the  top  of  the  tube 
containing  only  the  vapour  of  mercury. 

(4)  The   bulb   A    cannot   be   blown   of  exactly  the 
proper  capacity  for  giving  the  thermometer  its  required 
range.     For  this  reason,  it  is  at  first  made  too  large  and 
afterwards  reduced  in  the  following  manner. 

The  bulb  A  is  heated  to  a  temperature  shown 
by  a  thermometer  C,  which  is  already  finished, 
and  the  point  ra  to  which  the  mercury  rises 
in  the  stem  of  A  is  observed.  Next,  the 
bulb  A  is  heated  to  a  different  temperature, 
shown  by  the  thermometer  C,  and  the  point  n 
to  which  the  mercury  rises  in  the  stem  is 
again  observed.  Then,  knowing  the  differ- 
ence of  the  temperatures  shown  by  G,  and 
the  length  mn,  we  can  easily  calculate  how 
much  the  bulb  A  is  too  large.  Lastly,  the 
bulb  A  is  broken  off  to  the  required  size  and 
closed  up  again  without  admitting  air. 

When  the  thermometer  has  been  made  in 
the  manner  just  described,  it  is  found  that,  if  left  to  it- 
self, the  bulb  undergoes  a  gradual  contraction  which,  for 
practical  purposes,  may  be  considered  complete  in  six 
months.  We  ought,  therefore,  to  keep  the  thermometer 
at  least  six  months  before  proceeding  to  the  last  opera- 
tion. 

(5)  In  graduating  the  thermometer,  we  choose  two 


62  ELEMENTARY   THERMODYNAMICS. 

standard  temperatures  which  are  easily  reproduced  and 
can  be  proved  by  elaborate  experimental  methods  to 
remain  exactly  constant.  These  are  the  ultimate  tempe- 
rature assumed  by  a  mixture  of  ice  and  water  when  the 
only  external  influence  is  a  constant  pressure  of  one  atmo, 
and  the  temperature  of  the  steam  which  issues  from 
boiling  water  at  the  pressure  of  an  atmo. 

In  the  first  place,  when  the  pressure  of  the  air  is  one 
atmo,  the  thermometer  is  immersed  in  a  mixture  of  ice  and 
water  until  the  height  of  the  mercury  in  the  tube  has 
become  stationary ;  and  the  point  to  which  it  rises  is 
marked  by  a  scratch  on  the  glass,  known  as  the  freezing 
point.  Then  the  thermometer  is  surrounded  by  the  steam 
of  boiling  water  until  the  mercury  is  again  stationary, 
and  the  height  to  which  it  rises  in  the  tube  is  marked  by 
another  scratch,  called  the  boiling  point.  In  the  Centi- 
grade thermometer,  which  is  the  only  mercury  thermometer 
now  used  for  scientific  purposes,  these  two  temperatures 
are  marked  0  and  100.  The  portion  of  the  tube  between 
them  is  then  divided  into  100  parts  of  equal  capacity, 
each  of  which  is  called  a  degree,  and  the  divisions  are 
continued  beyond  the  two  standard  points  as  far  as  may 
be  required. 

It  is  necessary  that  all  thermometers  should  give  the 
same  indications  of  temperature,  not  merely  at  the  freezing 
and  boiling  points,  but  at  all  other  temperatures.  This 
condition  is  satisfied,  at  least  approximately,  by  making 
all  thermometers  of  the  same  kind  of  glass,  as  will  appear 
from  the  following  argument. 

If  we  take  a  given  mass  of  mercury,  or  of  glass  which 
is  not  very  irregular  in  form,  and  subject  it  to  no  stresses 
greater  than  the  pressure  of  the  air,  and  if  the  volume 


THE  CONSERVATION   OF  ENERGY.  63 

be  v0  when  a  standard  mercury  thermometer  is  at  0  and 
v  when  the  standard  thermometer  indicates  6,  it  is  found 
by  experiment  that 

v  =  v0  (1  +  a0), 

where  a  is  a  constant  number,  which,  for  mercury,  is 
about  -00018,  and  for  the  glass  of  thermometers,  '000025. 
Hence,  if  V  be  the  volume  of  the  mercury  and  s  the 
capacity  of  each  degree  of  the  stem  when  the  standard 
thermometer  is  at  zero,  and  if  the  mercury  rises  n  degrees 
in  the  stem  while  it  rises  6  in  the  standard  thermometer, 
the  volume  of  the  mercury  at  the  temperature  6  may  be 
written  in  either  of  the  forms 


or  (V+ns)(l+g0), 

where  m  =  '00018,  and  g  =  000025. 

We  have,  therefore,  very  approximately, 

V  (m-g)  6  =  ns. 

But  both  thermometers  are  marked  100  at  the  boiling 
point  :  hence 

V(m-g)  =  s, 

and  therefore  0  =  n, 

so  that  the  two  thermometers  agree. 

On  account  of  the  difficulty  of  obtaining  glass  perfectly 
homogeneous  and  of  exactly  a  certain  kind,  and  also  on 
account  of  the  irregular  form  of  the  thermometer  and  the 
internal  strains  which  are  produced  in  making  it,  it  is 
found  that  even  when  two  thermometers  have  been  made 
in  the  elaborate  manner  described  above,  their  readings 
do  not  generally  quite  agree  at  temperatures  distant  from 
the  boiling  and  freezing  points.  If  a  thermometer  is 


64  ELEMENTARY  THERMODYNAMICS. 

wanted  for  very  delicate  observations,  it  should  therefore 
be  compared  with  a  standard  thermometer. 

Two  other  systems  of  marking  the  thermometer  are 
often  used  for  non-scientific  purposes.  In  Fahrenheit's 
thermometer,  introduced  about  1714,  the  freezing  point 
is  marked  32  and  the  boiling  point  212.  In  Reaumur's, 
the  freezing  point  is  marked  0  and  the  boiling  point  80. 
In  stating  temperatures,  it  is  usual  to  indicate  the  scale 
referred  to  by  the  letters  C.,  F.,  R. 

27.  It  was  formerly  supposed  that  both  light  and 
heat  were  material  substances.  Light  was  believed  to 
consist  of  very  minute  bodies,  or  '  corpuscles,'  which  were 
shot  out  from  luminous  bodies  with  immense  rapidity  in 
straight  lines.  The  'corpuscular  theory/  which  is  prin- 
cipally due  to  Newton,  was  sufficient  to  explain  the 
commoner  properties  of  light.  But  a  different  theory, 
which  had  been  in  existence  before  Newton's  time,  was 
again  put  forward,  by  Dr  Young  in  England  about  the 
beginning  of  the  present  century,  and  in  France  by 
Fresnel  a  little  later.  Within  a  few  years  the  new 
doctrine,  according  to  which  light  consists  of  waves  of 
vibrating  motion,  was  so  completely  proved  that  it  became 
universally  accepted  by  1830. 

The  matter  which  was  supposed  to  form  heat  was 
called  'caloric,'  and  was  regarded  as  being  constant  in 
quantity,  like  matter  in  general.  According  to  this  view, 
the  conduction  of  heat  was  merely  the  transference  of 
caloric  out  of  one  body  into  another.  When  a  body 
expanded  through  rise  of  temperature,  the  increase  of 
volume  was  supposed  to  be  due  to  the  greater  amount 
of  caloric  present.  To  explain  the  increase  of  tempera- 


THE  CONSERVATION  OF  ENERGY.          65 

ture  that  occurs  when  a  body  is  suddenly  compressed,  it 
was  natural  to  suppose  that  caloric  was  squeezed  out  and 
so  rendered  sensible  to  the  thermometer  or  the  touch. 
Again,  it  was  supposed  that  different  bodies  required 
different  proportions  of  caloric  to  be  added  to  them  to 
produce  the  same  change  of  temperature :  this  was  ex- 
pressed briefly  by  saying  that  different  bodies  have  dif- 
ferent capacities  for  caloric ;  thus  a  pound  of  water  was 
considered  to  require  30  times  as  much  caloric  to  enter 
it  to  raise  its  temperature  by  1  °  as  a  pound  of  mercury. 

The  most  important  part  of  the  caloric  theory,  how- 
ever, was  the  doctrine  of  Latent  Heat,  propounded  by 
Dr  Black  in  1760.  When  it  was  found  that  by  applying 
heat  to  a  vessel  containing  ice,  no  change  of  temperature 
was  caused  so  long  as  the  contents  of  the  vessel  were 
kept  sufficiently  mixed,  until  the  whole  of  the  ice  was 
melted,  it  was  inferred  that  water  merely  differed  from  ice 
at  the  same  temperature  by  containing  a  much  larger 
quantity  of  caloric.  The  caloric  which  it  thus  appeared 
necessary  to  mix  with  ice  in  order  to  change  its  molecular 
state  without  altering  its  temperature,  was  called  '  latent,' 
because  it  could  not  be  detected  by  the  thermometer. 
In  like  manner,  the  steam  which  issued  from  boiling 
water  was  shown  by  the  thermometer  to  have  the  same 
temperature  as  the  water  itself,  and  was  therefore  sup- 
posed to  differ  from  it  only  by  containing  an  enormously 
greater  quantity  of  caloric. 

The  caloric  theory,  it  must  be  allowed,  afforded  a 
simple  explanation  of  the  phenomena  just  mentioned ; 
but  it  was  not  so  successful  in  accounting  for  the  heat 
developed  by  friction.  It  appeared  to  follow  from  the 
theory  that,  when  two  bodies  were  rubbed  together,  the 
P.  5 


66  ELEMENTARY   THERMODYNAMICS. 

friction  caused  a  diminution  in  the  capacity  of  one  or  both 
bodies  for  caloric,  in  consequence  of  which  the  caloric 
contained  within  them,  without  undergoing  any  increase, 
was  able  to  raise  them  to  a  higher  temperature ;  but  it 
was  only  necessary  to  carry  on  the  rubbing  process  long 
enough  to  show  that  the  quantity  of  heat  that  could  be 
produced  was  unlimited,  or  to  test  the  capacity  for  caloric 
of  each  body  after  being  rubbed,  to  see  that  this  view 
was  false. 

The  first  person  who  attained  a  correct  idea  of  the 
nature  of  heat  appears  to  have  been  Count  Rumford. 
Whilst  superintending  the  boring  of  cannon  in  the  arsenal 
at  Munich,  he  was  struck  with  the  enormous  quantity  of 
heat  produced  by  the  working  of  the  steel  borer,  and  in  a 
paper  which  he  published  on  the  subject  in  the  Philo- 
sophical Transactions  for  1798,  he  made  the  remarkable 
statement  that  the  source  of  the  heat  thus  developed 
appeared  to  be  inexhaustible.  He  then  observed  that 
since  heat  could  be  furnished  by  a  limited  system  in 
unlimited  amount,  it  could  not  be  a  material  substance. 
Following  up  this  idea,  he  argued  that  heat  was  motion 
and  made  the  first  attempt  to  calculate  the  quantity  of 
mechanical  work  that  must  be  expended  to  produce  a 
given  quantity  of  heat. 

In  the  year  following,  an  important  experiment  was 
described  by  Davy.  Two  pieces  of  ice  were  rubbed  to- 
gether until  both  were  nearly  melted  by  the  friction,  the 
water  thereby  produced  being  a  little  above  the  freezing 
point.  Here,  then,  it  was  clear  that  heat  was  actually 
created,  and,  therefore,  that  it  could  not  be  material. 
Also  as  the  capacity  of  water  for  heat  was  known  to  be 
much  greater  than  that  of  ice,  it  was  evident  that  friction 


THE  CONSERVATION  OF  ENERGY.          67 

did  not  diminish  the  capacities  of  bodies  for  heat  in  all 
cases,  as  the  caloric  theory  required  us  to  believe.  Davy 
was  thus  led  to  conclude  that  '  heat  may  be  denned  to  be 
a  peculiar  motion,  probably  a  vibration,  of  the  corpuscles 
of  bodies.' 

Yet  notwithstanding  these  decisive  experiments  and 
the  progress  of  the  new  theory  of  light,  the  doctrine  of 
caloric  continued  to  be  generally  adopted  until  about 
1840.  The  mechanical  theory  of  heat  was  then  revived 
almost  simultaneously  by  Mohr,  Seguin,  and  Mayer,  who 
based  their  ideas  on  theoretical  considerations,  and  by 
Colding  and  Joule,  who  appealed  to  experiment.  The 
eloquence  of  Mayer  caused  the  new  theory  to  be  generally 
assented  to:  the  numerous  and  brilliant  experiments  of 
Joule  proved  it  beyond  the  possibility  of  a  doubt. 

28.  In  one  of  Joule's  best  experiments,  water  was 
agitated  in  a  vessel  by  means  of  a  vertical  shaft  carrying 
a  number  of  paddles  which  worked  between  fixed  vanes, 
so  that  when  the  paddles  revolved,  the  water  was  pre- 
vented from  revolving  bodily  with  them,  and  in  consequence, 
it  offered  a  sufficient  resistance.  The  shaft  was  caused  to 
rotate  by  means  of  a  cord  wound  round  its  upper  part  and 
attached  to  a  heavy  weight  after  passing  over  a  pulley 
mounted  on  friction  wheels.  The  water  was  protected  from 
all  external  heating  and  other  effects,  except  the  pressure 
of  the  air,  and  the  weight  could  be  wound  up  without 
moving  the  paddles.  It  was  then  found,  after  making  all 
corrections,  that  the  work  required  to  be  done  by  the 
paddles  to  raise  the  temperature  of  the  water  by  any 
given  amount  was  always  proportional  to  the  quantity  of 
water.  In  addition  to  the  work  done  by  the  paddles, 

5—2 


68  ELEMENTARY   THERMODYNAMICS. 

there  will  be  a  very  small  amount  of  work  done  by  the 
pressure   of  the   air,  which,  for   given   initial   and   final 


temperatures,  will  also  be  proportional  to  the  quantity  of 
water,  since,  for  a  short  time,  the  pressure  of  the  air  may 
be  considered  constant.  If,  for  the  sake  of  numerical 
illustration,  we  suppose  the  mass  of  water  to  be  a  gramme 
and  the  pressure  of  the  air  exactly  an  atmo,  the  work 
done  by  the  paddles,  when  the  temperature  of  the  water 
rises  from  0°  C.  to  1°  C.,  is  found  to  be  over  40  million 
ergs ;  and  since  the  volume  of  the  water  at  1°  C.  is  less 
than  the  volume  at  0°  C.  by  less  than  the  10,000th  of  a 
cubic  centimetre,  the  work  done  on  the  water  by  the 
pressure  of  the  air  will  be  less  than  101 '3  ergs. 

Experiments  were  also  made  by  Joule  on  other  sub- 
stances with  a  like  result.  Again,  in  a  number  of  other 
experiments,  heat  was  produced  indirectly  by  expending 
mechanical  work  on  a  system  so  as  to  cause  its  parts  to 


THE  CONSERVATION  OF  ENERGY.          69 

become  electrified,  and  then  allowing  them  to  resume 
their  unelectrified  state  ;  and  it  was  found  that  the  final 
effect  was  the  same  as  if  the  same  amount  of  work  had 
been  expended  directly.  Also  it  was  shown  that  when 
•heat  is  developed  by  chemical  action,  the  ultimate  result 
is  the  same  whether  the  change  takes  place  by  means  of 
electric  agency  or  otherwise. 

From  these  and  other  experiments  it  is  concluded  that, 
however  simple  or  complicated  may  be  the  electric  and 
other  properties  of  any  system,  or  whatever  may  be  the 
mechanical  motions,  if  it  be  protected  from  all  external 
heating  and  electric  effects,  the  work  that  must  be  done 
on  it  to  bring  it  from  any  given  state  P  to  any  other 
given  state  Q  depends  only  on  those  states  and  not  on  the 
particular  manner  in  which  the  change  of  state  is  pro- 
duced. To  put  this  statement  into  a  simple  form,  let  us 
choose  a  standard  state  0  and  denote  the  work  required 
to  bring  the  system  from  the  state  0  to  the  state  P  by  U. 
Then  so  long  as  we  keep  to  the  same  standard  state,  U 
will  depend  on  the  state  P  only  and  may  be  written  Up. 
Also  by  supposing  the  system  to  be  brought  from  the  state 
0  to  the  state  Q  by  first  bringing  it  from  0  to  P  and  then 
from  P  to  Q,  we  see  that 

0,  -&>«„, 

where  wpq  is  the  work  required  to  bring  the  system  from 
Pto  Q. 

Again,  if  a  different  standard  state  0'  be  chosen  and  the 
new  value  of  U  be  denoted  by  U',  we  shall  have 


so  that  the  value  of  U  is  only  altered  by  a  constant. 


70  ELEMENTARY   THERMODYNAMICS. 

Hence,  the  quantity  U  is  a  single-valued  function  of 
the  independent  variables  which  define  the  state  of  the 
system,  together  with  an  arbitrary  additive  constant  de- 
pending only  on  the  choice  of  a  standard  state,  and  dw, 
the  work  done  on  the  system  in  any  small  change  of  state, 
is  an  exact  differential  given  by  dU=dw. 
The  quantity  U  is  called  the  energy  of  the  system,  but 
it  should  be  noticed  that  the  present  definition  is  not 
quite  the  same  as  that  given  previously.  In  fact,  the 
values  of  U,  according  to  the  two  definitions,  differ  by  a 
constant ;  for  the  standard  state  in  the  former  part  of  the 
chapter  did  not  make  the  whole  energy  zero,  but  only  that 
part  of  it  which  we  called  potential  energy. 
If  the  system  consists  of  a  number  of  bodies  which 
exert  forces  on  one  another  without  being  actually  in 
contact,  as  in  the  case  of  gravitation,  it  is  obvious  that 
the  energy  of  the  system  will  depend,  not  only  on  the 
state  of  each  of  its  parts,  but  also  on  their  relative  posi- 
tions with  respect  to  one  another.  That  part  of  the 
energy  which  depends  on  the  relative  positions  of  the 
bodies  is  called  their  '  mutual  energy,'  while  the  energy 
of  any  particular  body  may  be  distinguished  as  its  '  in- 
ternal energy.' 

29.  Again,  if  a  system  be  protected  from  external 
electric  influences  only,  it  is  found  that  no  change  of  state 
can  be  produced  by  the  joint  agency  of  heat  and  work 
which  cannot  be  produced  by  work  alone,  and  therefore 
we  may  always  define  the  energy  of  the  system  in  any 
state  P,  into  which  it  can  be  brought  by  the  combined 
effects  of  heat  and  work,  to  be  the  work  that  must  be 
done  on  the  system  to  bring  it  from  the  standard  fixed 


THE   CONSERVATION   OF   ENERGY.  71 

state  0  to  the  given  state  P  on  the  supposition  that  both 
external  heating  and  external  electric,  influences  are 
absent.  Also,  since  it  appears  from  experiment  that  in 
any  change  of  state  due  to  the  joint  action  of  heat  and 
work,  the  increase  of  the  energy  is  not  generally  the  same 
as  the  work  actually  done  on  the  system,  we  shall  define 
the  heat  absorbed  by  the  system  in  any  change  of  state  in 
which  there  are  no  external  electric  influences,  to  be  the 
algebraic  excess  of  the  increase  of  the  energy  over  the 
work  done  on  the  system.  If  the  change  of  state  be  in- 
definitely small,  and  if  dU  be  the  increase  of  energy,  dW 
the  work  done  on  the  system,  and  dQ  the  heat  absorbed, 
we  shall  therefore  have 


Again,  since  a  change  of  state  which  can  be  produced  by 
heat  alone  can  also  be  produced  by  work  alone,  or  partly  by 
heat  and  partly  by  work,  in  an  infinite  number  of  ways,  it 
is  evident  that  in  any  change  of  state  due  to  the  action  of 
heat  and  work,  the  work  done  on  the  system  and  the  heat 
absorbed  by  it  will  generally  depend  on  the  way  in  which 
the  change  of  state  takes  place  as  well  as  on  the  initial 
and  final  states  themselves.     In  mathematical  language, 
the  quantities  d  W  and  dQ,  though  functions  of  the  inde- 
pendent variables  (x,y,z,  ......  )  which  define  the  state  of 

the  system,  and  of  their  differentials,  are  not  generally 
complete  differentials  of  functions  of  (x,y,z,  ......  ). 

The  word  '  heat,'  in  the  popular  sense,  refers  to  some- 
thing which  exists  in  a  body,  and  therefore  in  any  change 
of  state,  its  increase  depends  only  on  the  initial  and  final 
states  and  not  on  the  way  in  which  the  change  of  state 
takes  place.  The  popular  meaning  of  the  word  '  heat  '  is 


72  ELEMENTARY   THERMODYNAMICS. 

therefore  different  from  that  which  we  have  assigned  to 
it.  In  fact,  according  to  the  popular  usage,  '  heat '  means 
non-mechanical  kinetic  energy,  while  our  definition  makes 
it  identical  with  non-mechanical  work;  non-mechanical  work 
done  on  the  system  being  the  same  as  heat  absorbed,  and 
non-mechanical  work  done  by  the  system  against  the  ex- 
ternal forces,  the  same  as  heat  given  out.  To  prevent 
confusion,  we  shall  never  use  the  word  'heat'  in  the 
popular  sense,  but  always  speak  of  non-mechanical  kinetic 
energy. 

Heat  is  absorbed  either  by  conduction  or  radiation. 
Conduction  is  a  surface  phenomenon  which  takes  place 
when  bodies  are  actually  in  contact  and  is  merely  the 
transference  of  energy  out  of  one  body  into  another: 
radiation  is  an  exchange  of  energy  between  a  material 
body  and  the  ether,  and  may  occur  in  the  interior  of  the 
body.  A  common  form  of  conduction  is  known  as  con- 
vection and  may  be  observed  when  a  pan  of  cold  water  is 
set  on  the  fire ;  a  continual  circulation  being  kept  up  by 
the  descent  of  the  colder  water  from  above  to  take  the 
place  of  the  water  which  ascends  after  being  warmed  and 
made  lighter  by  contact  with  the  heated  bottom  of  the 
pan.  In  the  case  of  the  sun,  an  immense  flood  of  light 
and  heat  is  radiated  from  the  glowing  interior  through 
the  enormous  atmosphere  of  cooler  gases  with  which  it  is 
surrounded. 

The  principle  of  the  equivalence  of  heat  and  work,  as 
expressed  by  the  equation  dU=dW  +  dQ,  is  known  as  the 
first  Law  of  Thermodynamics.  It  is  a  particular  case  of 
the  principle  of  the  Conservation  of  Energy.  The  second 
Law  of  Thermodynamics  is  Carnot's  axiom,  which  is 
merely  a  law  of  friction. 


THE  CONSERVATION  OF  ENERGY.          73 

30.  If  a  system  be  exposed  to  external  electric 
influences,  it  is  found  that  it  may  acquire  such  electric 
properties  by  contact  with  other  systems  that  it  may  be 
impossible  to  bring  it  back  to  the  standard  fixed  state 
until  the  new  electric  properties  are  given  back.  We  are 
then  to  regard  the  system  as  a  new  system  and  to  choose 
a  new  state  as  the  standard  state  from  which  to  reckon 
the  energy.  Again,  when  the  different  states  of  the 
system  are  comparable  with  the  same  standard  state,  it  is 
found  that,  if  there  are  electrified  or  magnetized  bodies  in 
the  neighbourhood,  the  energy  of  the  system  may  be 
altered  without  doing  work  on  it,  even  when  it  is  evident 
that  there  are  no  external  heating  influences.  The  system 
is  then  popularly  said  to  absorb  'electric  energy/  the 
meaning  of  which  is  fully  explained  in  books  on  electricity 
and  magnetism.  In  general,  if  dU  be  the  increase  of 
energy,  dW  the  work  done  on  the  system,  and  dE  the 
'  electric  energy  '  absorbed,  or  the  '  electric  work  '  done  on 
the  system,  calculated  as  in  works  on  electricity,  the  heat 
absorbed,  dQ,  is  defined  to  be  the  value  ofdU-dW-dE, 
so  that  the  principle  of  energy  may  be  written  in  the 
form 


As  we  do  not  wish  to  enter  into  the  details  of 
electricity  and  magnetism,  we  shall,  in  future,  always 
suppose  our  system,  whether  electrified  or  not,  to  be  so 
chosen  that  there  are  no  external  electric  influences, 
except  in  a  few  special  cases.  The  principle  of  energy 
then  takes  the  simple  form 


where  dW  and  dQ  are  not  generally  perfect  differentials. 


74  ELEMENTARY   THERMODYNAMICS. 

31.     The  equation 

dU=dW+dQ 
may  be  written 

dQ  =  dU-dW. 

Hence  if  we  suppose  the  body  or  system  of  bodies  to 
undergo  a  finite  change  of  state,  in  which  U  changes  from 
U0  to  f/i,  we  have 

fdQ=Ul-U0-JdW. 

If  the  changes  of  state  be  such  that  the  final  state  is  the 
same  as  the  first,  the  series  of  operations  which  the  system 
goes  through  is  called  a  cyclical  process.  We  have  then 
Ui=U0t  so  that 


Now  —  fd  W  is  the  external  work  done  by  the  system  : 
hence  when  a  system  which  is  protected  from  external 
electric  influences  undergoes  a  cyclical  process,  the  heat 
absorbed  is  exactly  equivalent  to  the  work  done  by  the 
system. 

32.  The  result  that  dW  and  dQ  are  not  generally 
perfect  differentials  is  so  important  that  it  will  be  advan- 
tageous to  consider  some  simple  illustrations. 

Suppose,  in  the  first  place,  that  a  quantity  of  water 
and  steam  at  a  high  temperature,  and  therefore  also  at  a 
high  pressure,  is  contained  in  a  cylinder  fitted  with  a 
smooth  air-tight  piston,  and  let  it  be  required  to  reduce 
the  contents  of  the  cylinder  to  a  much  smaller  volume 
and  a  lower  temperature,  say  the  ordinary  freezing  point  ; 
suppose  also  that  the  final  volume  is  too  great  to  be  filled 
by  the  water  within  the  cylinder,  so  that  steam  must  also 
be  present  and  the  final  pressure  inside  the  cylinder  be 
consequently  very  small. 


THE  CONSERVATION  OF  ENERGY.          75 

The  desired  change  of  state  may  be  brought  about  in 
many  different  ways,  of  which  two  will  be  here  considered. 

(1)  Let  the  piston  be  forced  in  so  as  to  reduce  the 
volume  of  the  interior  of  the   cylinder  to  the  required 
extent  before  any  heat  is  abstracted,  and  then  reduce  the 
temperature  to  its  final  value  without  altering  the  volume. 
The  work  done  on   the  piston  in    this   method  will   be 
considerable. 

(2)  First  suppose  heat   to  be   abstracted   from   the 
cylinder,  without  altering  the  volume,  until  the  desired 
lower  temperature  is  attained.     Then  let  the  piston  be 
forced  in  while  the  temperature  of  the  interior  of  the 
cylinder   is   kept    constant   by   the   conduction   of    heat 
through  its  sides.     The  work  done  in  this  method  will  be 
negligible,  since  the  pressure  against  which  the  piston  has 
to  be  forced  is  very  small. 

Since  the  work  done  on  the  system  while  it  ex- 
periences a  given  change  of  state  depends  on  the  way  in 
which  the  change  of  state  takes  place,  it  follows  that  dW 
and  dQ  are  not  complete  differentials,  like  dU. 

Again,  consider  the  following  example.  A  cylinder 
fitted  with  a  smooth  air-tight  piston  contains  a  quantity 
of  air  at  any  given  temperature  and  at  volume  v,  and  it  is 
required  to  increase  the  volume  to  v'  without  altering  the 
temperature. 

We  will  show  by  two  illustrations  how  this  may  be 
done  in  different  ways. 

(1)  Let  the  piston  be  drawn  out  so  slowly  that  the 
air  within  is  able  to  exert  its  maximum  pressure  on  it  and 
the  temperature  to  be  kept  constant  by  the  conduction  of 
heat  through  the  sides  of  the  cylinder. 

(2)  Again,  suppose  the  piston  drawn  out  so  rapidly 


76  ELEMENTARY   THERMODYNAMICS. 

that  the  air  within  is  hardly  able  to  keep  tip  with  it  and 
therefore  unable  to  do  much  work,  and  then  let  the 
temperature  be  brought  to  its  former  value. 

The  work  done  on  the  piston  by  the  contents  of  the 
cylinder  being  clearly  different  in  the  two  cases,  we  are 
led  to  the  same  conclusions  with  respect  to  d  W  and  dQ 
as  before. 

33.  If  a  body  be  subjected  to  a  uniform  normal 
pressure  p,  the  work  done  on  it  during  a  small  increase  of 
volume  dv  is  -pdv.  The  equation  dU=  dW  +  dQ  then 
becomes  dQ  =  d  U  +  pdv.  Hence  if  either  p  or  v  be  con- 
stant, the  heat  absorbed  by  the  body  in  any  change  of 
state  will  depend  only  on  the  initial  and  final  states.  We 
are  accordingly  led  to  the  following  definitions,  in  which 
the  state  of  the  body  is  supposed  to  depend  only  on  the 
temperature  when  either  p  or  v  is  given — a  supposition 
which  requires  that  there  should  be  neither  electric 
actions  nor  mechanical  motions: — 

(1)  At  any  temperature,  the  '  thermal  capacity  of  a 
body  at  constant  pressure '  is  the  heat  required  to  raise 
its  temperature  one  degree  C.  while  the  pressure  remains 
constant. 

The  pressure  which  most  frequently  occurs  is  that  of 
the  air,  which,  at  any  instant,  may  be  supposed  to  remain 
constant  for  a  short  time. 

(2)  At  any  temperature,  the  'thermal  capacity  of  a 
body  at  constant  volume '  is  the  heat  required  to  raise  its 
temperature   one   degree    C.  while   the   volume   remains 
constant. 

If  the  body  is  homogeneous,  the  thermal  capacity  of  a 
mass  of  one  gramme  is  called  its  '  specific  heat.' 


THE  CONSERVATION  OF  ENERGY.          77 

The  specific  heat  of  water  at  0°  C.  under  a  constant 
pressure  of  one  atmo  is  called  a  Calorie,  and  is  used  as  an 
arbitrary  unit  of  heat. 

34.  In  Joule's  experiments  on  the  agitation  of 
water,  the  apparatus  cannot  be  arranged  so  that  the  water- 
rises  in  temperature  exactly  from  0°  C.  to  1°  C.  In  order 
to  find  the  mechanical  value  of  a  calorie,  we  must  there- 
fore determine  the  specific  heats  of  water  at  different 
temperatures.  This  may  be  done  by  a  method  known  as 
the  'method  of  mixtures.'  Thus  let  a  quantity  of  water 
at  0°  C.  be  added  to  an  equal  quantity  at  some  other 
temperature,  say  100°  C.,  and  let  the  mixture  be  protected 
from  all  external  influences  except  the  pressure  of  the  air, 
which  may  be  taken  to  be  an  atmo.  Then  if  the  uniform 
temperature  which  the  water  finally  assumes  be  #C.,  and 
if  no  appreciable  amount  of  work  has  been  done  either  by 
gravity  or  in  mixing  them  together,  it  follows  that,  since 
the  work  done  by  the  pressure  of  the  air  is  negligible,  the 
heat  required  to  raise  the  temperature  of  any  mass  of 
water  from  0°  C.  to  #C.,  under  a  constant  pressure  of  one 
atmo,  is  equal  to  that  required  to  raise  the  temperature 
of  an  equal  mass  of  water  from  6C.  to  100°  C.  under  the 
same  pressure.  In  this  way,  the  specific  heats  of  water 
under  a  constant  pressure  of  one  atmo  have  been  deter- 
mined in  calories  at  different  temperatures,  thus : — 
at  0CC., 1-000 

...  10°  C., 1-0005 

...  20°  C., 1-0012 

...  30°  C., 1-0020. 

By  means  of  these  results,  it  is  found  from  Joule's 


78  ELEMENTARY   THERMODYNAMICS. 

experiments  that  a  calorie  is  equivalent  to  42350  gramme- 
centimetres,  or  41,539,759-8  ergs,  or  about  3  foot-pounds. 
In  English  measure,  the  heat  required  to  raise  the 
temperature  of  1  Ib.  of  water  from  0°  C.  to  1°  C.  under  a 
pressure  of  one  atmo  is  equivalent  to  about  1390  foot- 
pounds. 

Now  if  a  mass  of  one  gramme  be  moving  without 
rotation  or  vibration  at  a  speed  of  v  centimetres  per 
second,  its  mechanical  kinetic  energy  will  be  \vz  ergs.  If 
this  be  equivalent  to  a  calorie,  we  shall  have 

tf  =  83,079,519-6, 
or  0=9114-8. 

Thus  the  velocity  must  be  91  "148  metres,  or  299  feet,  per 
second,  that  is,  5'469  kilometres,  or  3'4  miles,  per  minute. 
Hence  if  two  equal  masses  of  water  at  0°  C.  be  moving 
with  this  velocity  and  impinge  on  one  another  in  such  a 
way  that  they  are  both  brought  to  rest ;  then  if  no  steam 
be  formed,  the  impact  will  be  sufficient  to  raise  the 
temperature  by  1°  C.  For  a  rise  from  0°  C.  to  100°  C.,  a 
velocity  of  about  55  kilometres,  or  34  miles,  per  minute 
is  required.  In  the  case  of  iron,  the  specific  heat  is  only 
about  £  of  a  calorie,  and  therefore  the  velocities  are  only 
about  |  as  large  as  for  water. 

Again,  let  a;  be  the  latent  heat  of  liquefaction  of  ice, 
in  calories,  under  a  pressure  of  one  atmo,  that  is,  the 
number  of  calories  required  to  convert  one  gramme  of  ice 
at  Oc  C.  into  water  at  the  same  temperature.  Also 
suppose  that  i  grammes  of  ice  at  0°  C.  are  mixed  with  w 
grammes  of  water  at  0C.,  and  that,  in  consequence,  the 
whole  of  the  ice  is  melted  under  a  constant  pressure  of 
one  atmo.  Then  if  no  heat  be  allowed  to  escape,  and  if 


THE  CONSERVATION  OF  ENERGY. 


79 


O'C.  be  the  final  temperature,  the   heat   gained  by  the 
water  from  the  ice  will  be,  very  approximately,  w  (6'  —  0) 
calories,  and  the  heat  gained  by  the  ice  from  the  water, 
i  (x  +  6'}  calories.     Hence,  very  approximately, 
w  (6'  -0)  +  i  (x  -f  &}  =  0, 

wO     iv  +  i  „, 

or  x=— -6 . 

%  i 

It  is  thus  found  that  the  latent  heat  of  ice  under  a 
pressure  of  one  atmo  is  79*25  calories,  or  3,292,025,964 
ergs. 

Bunsen's  calorimeter  is  a  small  instrument  by  which 
the  heat  coming  from  a  small  solid  body  under  the  constant 
pressure  of  the  air  is  easily  and  accurately  determined  by 
the  melting  of  ice.  It  consists  of  the  three  parts,  a,  b,  c, 


made  of  glass,  and  sealed  together  with  the  blow-pipe. 
The  part  b  contains  distilled  water  freed  from  air  by 
boiling,  and  the  bottom  of  b  and  the  tube  c  are  filled  with 
boiled  mercury,  the  upper  part  of  the  tube  being  bent 
horizontally,  calibrated,  and  graduated.  In  preparing  the 
calorimeter  for  use,  a  coating  of  ice  is  formed  round  the 
test  tube  a  by  passing  a  stream  of  alcohol,  previously 


80  ELEMENTARY  THERMODYNAMICS. 

cooled  below  0°  C.  in  a  freezing  mixture,  down  to  the 
bottom  of  a  and  back  again.  The  calorimeter  is  then 
placed  in  a  vessel  filled  with  clean  snow,  a  substance 
which  soon  acquires  and  long  maintains  a  temperature 
0°  C.,  unless  the  temperature  of  the  room  is  below  0°  C. 
Lastly,  the  test  tube  a  is  partially  filled  with  water  or 
some  other  fluid  which  does  not  dissolve  the  body  to  be 
experimented  on,  and  as  soon  as  the  whole  is  at  the 
temperature  0°  C.,  the  calorimeter  is  ready  for  use. 

In  making  an  experiment,  the  body  which  is  to  give 
off  the  heat  is  dropped  into  the  test  tube  a.  This  will 
cause  the  water  in  a  to  become  warmer,  and  then,  by  the 
conduction  of  heat  through  its  sides,  some  of  the  ice  which 
surrounds  it  will  be  melted.  This  will  go  on  till  the 
temperature  of  the  whole  is  again  reduced  to  0°  C.  If 
n  grammes  of  ice  be  melted  in  the  process,  the  heat  given 
off  by  the  body  will  be  79'25  calories.  The  value  of  n  is 
determined  by  the  movement  of  the  mercury  in  the 
graduated  tube,  depending  on  the  fact  that  at  a  pressure 
of  one  atmo  and  at  0°  C.,  one  gramme  of  ice  occupies 
1-087  cubic  centimetres  and  one  gramme  of  water,  only 
1-00011.  It  should  be  observed  that  if  any  air  be  allowed 
to  remain  in  the  water  in  6,  it  will  be  expelled  in  the 
form  of  a  small  bubble  during  the  process  of  freezing  the 
water  around  the  test  tube  a,  and  partially  re-dissolved 
when  the  ice  is  melted.  A  small  error  will  thus  be  intro- 
duced into  the  indications  of  the  calorimeter. 


CHAPTER  II. 

ON   PERFECT   GASES. 

35.  WE  are  unable  to  proceed  much  further  with  the 
first  law  of  thermodynamics  until  we  have  introduced 
Carnot's  principle,  except  in  the  case  of  the  more  per- 
manent gases,  where  some  simple  experiments  supply  us 
with  the  information  we  require.  The  most  common  of 
these  gases  are  Air,  Oxygen,  Hydrogen,  Nitrogen,  Car- 
bonic Oxide,  Carbonic  Acid,  Chlorine,  Cyanogen,  Marsh 
Gas,  Olefiant  Gas,  Sulphurous  Acid,  and  Ammonia.  They 
are  often  called  '  perfect '  gases,  because  they  all  exhibit 
the  same  simple  properties  and  obey  the  same  laws  more 
or  less  perfectly ;  but  the  first  five  and  Marsh  Gas  are 
more  perfect  than  the  others. 

We  shall  suppose  the  gas  contained  in  a  closed  vessel 
which  is  in  a  state  of  mechanical  rest  and  either  main- 
tained at  a  uniform  temperature  or  prevented  from 
receiving  or  losing  heat  by  being  wrapped  up  in  some 
non-conducting  material,  like  felt.  For  simplicity,  we 
shall  also  suppose  that  electric  and  magnetic  actions  are 
entirely  absent.  Under  these  conditions  it  is  found  that 
the  gas  quickly  assumes  a  state  of  equilibrium  in  which 
p.  6 


82 


ELEMENTARY   THERMODYNAMICS. 


the  temperature  has  the  same  value  in  every  part  of  the 
vessel.  It  is  also  found  that  the  gas  exerts  a  normal 
pressure  against  the  interior  of  the  containing  vessel ;  and 
that  the  pressure  of  the  gas,  also  the  'density,'  that  is, 
the  mass  per  unit  volume,  and  the  '  specific  volume,'  that 
is,  the  volume  per  unit  mass,  have  the  same  values  at  all 
points  in  the  same  horizontal  plane.  Owing  to  gravity,  the 
density  is  not  quite  the  same  at  points  not  in  the  same 
horizontal,  but  the  difference  is  too  small  to  be  taken  into 
account  in  the  present  chapter.  If  gravity  be  neglected, 
the  density  of  the  gas  will  have  the  same  value  throughout 
the  vessel  and  the  pressure  the  same  value  all  over  the 
surface.  These  values  may  then  be  referred  to  briefly  as 
the  density  and  pressure  of  the  gas. 

36.  A  very  important  experiment  on  perfect  gases, 
first  made  by  Gay  Lussac,  was  repeated  by  Joule  in  1844 
in  a  greatly  improved  form.  In  -a  vessel  of  water  A,  there 


were  two  strong  closed  vessels  B,  C,  connected  by  a  pipe 
m  which  there  was  a  very  perfect  stop-cock.  Air  was 
compressed  to  a  pressure  of  about  20  atmospheres  in  B 
and  exhausted  from  (7,  and  the  temperature  of  the  water 


ON   PERFECT  GASES.  83 

in  A  was  made  uniform  by  stirring.  The  stop-cock  was 
then  opened,  so  that  the  air  rushed  from  B  to  C,  and,  in 
consequence,  the  water  near  B  was  cooled  while  that  near 
C  was  heated.  The  water  in  A  being  again  brought  to  a 
uniform  temperature  by  stirring  and  the  proper  correction 
made  for  the  work  thus  expended  as  well  as  for  the  heat 
lost  from  A  by  conduction  or  radiation,  it  was  found  that 
no  perceptible  change  of  temperature  had  been  produced 
in  the  water  by  the  expansion  of  the  air.  It  was  therefore 
inferred  that,  on  the  whole,  no  heat  had  been  gained  or 
lost  by  the  air  by  means  of  the  water  during  the  opera- 
tion ;  and  as  no  external  work  had  been  done  by  it,  it 
followed  from  the  first  law  of  thermodynamics,  that  the 
energy  was  unaffected  by  the  process.  In  other  words, 
the  energy  of  the  air  is  constant  so  long  as  the  tempe- 
rature is  constant.  Hence  we  conclude  that  the  energy 
of  a  given  mass  of  air  in  a  state  of  equilibrium  is  a 
function  of  the  temperature  only. 

This  remarkable  experimental  result  was  tested,  both 
for  air  and  some  other  gases,  by  Joule  and  Thomson  a  few 
years  later  in  a  series  of  careful  experiments,  and  was 
found  to  be  very  approximately  correct  in  every  case.  In 
consequence,  it  is  generally  taken  to  be  true  for  all  the 
more  permanent  gases. 

Hence,  if  a  quantity  of  any  perfect  gas  expand  in  any 
way  from  one  volume  to  another,  and  the  temperature  be 
made  the  same  after  the  expansion  as  before,  the  energy 
will  be  unaltered  by  the  operation,  and  therefore,  by  the 
first  law  of  thermodynamics,  the  heat  absorbed  by  the 
gas  (in  ergs)  will  be  equal  to  the  work  done  by  it  during 
the  expansion. 

Again,  if  the  temperature  of  a  given  mass  of  gas  be 

6—2 


84  ELEMENTARY   THERMODYNAMICS. 

raised  by  any  amount  while  the  volume  remains  constant, 
the  increase  of  energy  will  depend  only  on  the  initial  and 
final  temperatures,  and  no  work  being  done  by  the  gas, 
the  same  will  also  be  true  of  the  heat  absorbed.  But  the 
heat  absorbed  by  one  gramme  of  any  substance  when  the 
temperature  is  raised  1°  C.  and  the  volume  kept  constant, 
is  the  specific  heat  of  the  substance  at  constant  volume. 
In  the  case  of  a  perfect  gas,  this  is  therefore  a  function  of 
the  temperature  only,  or  a  constant. 

37.  There  is  a  remarkable  relation  between  the 
pressure,  density,  and  temperature  of  a  perfect  gas. 
Suppose,  for  example,  that  a  gramme  of  any  perfect  gas  is 
contained  in  a  cylinder  fitted  with  a  smooth  air-tight 
piston,  so  that  the  volume  of  the  gas  can  be  increased  or 
decreased  at  pleasure.  Then  it  is  found  that,  so  long  as 
the  temperature  of  the  gas  is  kept  constant,  the  pressure 
varies  inversely  as  the  volume.  The  pressure  of  the  gas 
at  any  instant  being  denoted  by  p  and  the  volume  by  v, 
it  follows  that  the  product  pv  depends  only  on  the 
temperature.  If  we  denote  the  temperature,  as  indicated 
by  the  mercury  centigrade  thermometer  by  6',  we  may 
therefore  write 

P>=/W (12). 

Again,  if  the  pressure  be  kept  constantly  equal  to  any 
value  p,  it  is  found  that  the  changes  of  volume  due  to 
changes  of  temperature,  may  be  expressed  by  the  formula 

v=V(l+a8') (13), 

where  F  is  the  volume  when  &  =  0  and  a  has  the  same 
value  -003665,  or  ^,  not  only  for  all  pressures,  but  for 
all  gases. 


ON   PERFECT   GASES.  85 

The  former  of  these  laws  was  discovered  by  Boyle, 
the  latter  by  Charles.  By  continental  writers  they  are 
generally  referred  to  as  the  laws  of  Mariotte  and  Gay 
Lussac.  It  has  been  shown  by  Regnault  that  they  are 
not  strictly  accurate,  but  the  deviations  appear  to  be  very 
small  when  the  gas  is  sufficiently  removed  from  its  point 
of  condensation,  that  is,  when  the  pressure  is  not  too 
great  or  the  temperature  too  low. 

From  equation  (12)  we  obtain 


and  therefore,  by  equation  (13), 
that  is, 


where  k  is  a  constant  depending  only  on  the  nature  of 
the  gas. 

If  we  write  0  for  273  +  ff  and  R  for  ^  ,  we  get 

pv  =  R9  ........................  (14), 

and  therefore  if  (pQ,  v0,  00)  be  any  corresponding  values 
of  (p,  v,  6\ 

7=^  ........................  (15). 

The  result  pv  =  R6  may  be  shown,  by  Carnot's  prin- 
ciple, to  involve  the  experimental  fact  that  the  energy 
depends  only  on  the  temperature. 

38.  The  simple  relation  expressed  by  the  equation 
pv  =  Rd  for  any  perfect  gas  has  led  to  the  construction  of 
thermometers  in  which  air  is  employed  as  the  thermo- 


86  ELEMENTARY  THERMODYNAMICS. 

metric  substance  instead  of  mercury.  In  these  thermo- 
meters, either  the  pressure  or  the  volume  is  kept  constant. 
In  both  kinds,  6  is  taken  to  be  273  at  0°  C.,  and  R  is 
determined  from  the  equation  pv  =  R6  by  observing  the 
volume  at  0°  C.  corresponding  to  the  given  pressure,  or 
the  pressure  at  0°  C.  corresponding  to  the  given  volume, 
At  any  other  temperature,  the  temperature  of  the  air 

thermometer  is  denned  to  be  the  value  of  ^ ,  where,  in 

M 

constant  pressure  thermometers,  the  value  of  v  is  to  be 
found  by  experiment,  and  in  constant  volume  thermo- 
meters, the  value  of  p.  The  indications  of  constant 
pressure  and  constant  volume  thermometers  agree  with 
one  another,  but  they  are  not  quite  the  same  as  those  of 
the  common  mercury  thermometer  increased  by  273. 
Since  any  of  the  perfect  gases  may  be  used  instead  of  air, 
the  scale  of  the  air  thermometer  is  often,  but  improperly, 
called  '  absolute.'  A  truly  '  absolute '  scale  of  tempe- 
rature, independent  of  the  special  properties  of  any 
particular  substance  or  class  of  substances,  will  be  ob- 
tained in  the  next  chapter  by  means  of  Carnot's  principle, 
and  it  will  be  found  to  coincide  practically  with  the  scale 
of  the  air  thermometer. 

In  a  very  simple  form  of  the  air  thermometer,  due  to 
Jolly,  the  volume  of  the  air  is  not  kept  quite  constant, 
but  the  only  change  in  it  is  that  due  to  the  small  ex- 
pansion by  heat  of  the  vessel  in  which  it  is  contained. 
This  thermometer  consists  of  a  glass  globe  of  about  50 
cubic  centimetres  capacity  formed  in  one  piece  with  a  fine 
capillary  bent  tube  A  and  a  larger  tube  R  The  glass 
globe  is  filled  with  dry  air  and  the  tube  B  is  joined  to 
an  open  glass  tube  G  by  an  india-rubber  tube  D;  the 


ON    PERFECT  GASES. 


87 


EC 


tubes  B,  D,  and   the    lower  part  of  C  being  filled  with 
mercury. 

In  graduating  the  instrument,  we  proceed  as  follows : — 

(1)  The  glass  globe  is  surrounded  with  melting  ice 
to  bring  its  temperature  to  0°  C.     The  tube  G  is  then 
raised  or  lowered  till  the  surface  of  the  mercury  in  B  is 
brought  to  a  mark  near  the  capillary  tube  A.     Lastly, 
the  pressure  p0,  in  the  glass  globe,  is  given  by  the  height 
of  the  barometer  and  the  height  of  the  mercury  in  C 
above  the  mark  on  B. 

This  operation  is  equivalent  to  finding  the  value  of  R. 

(2)  The  glass  globe  is  then  brought  to  any  other 
temperature  we  wish  to  determine,  the  mercury  adjusted 
and  the  pressure  p  obtained,  as  before. 

If  V  be  the  capacity  of  the  glass  globe  at  0°  C.,  its 
capacity  when  the  temperature  of  the  air  thermometer 
is  0,  will  be 

V  (1  + -000025  (6  -  273)}. 

Hence  since  the  quantity  of  air  contained  in  the  capillary 
tube  is  negligible,  we  shall  have 

pV  [1  +  -000025  (6  -  273)}  =  R0, 


88 

and 

Thus 


whence 


ELEMENTARY    THERMODYNAMICS. 


8  -  273  = 


•000025  (6  -  273)), 
1  +  -000025  £-(8-  273), 

Po 
P-Po 


•003665p0  -  -000025p ' 

by  means  of  which  6  is  found  from  the  observed  value 
of  p. 

39.  The  following  table1  exhibits  some  important 
fundamental  experimental  results  relating  to  the  perfect 
gases,  at  a  pressure  of  one  atmo  and  at  0°  C. 


Air 
Oxygen  (0) 
Hydrogen  (H) 
Nitrogen  (N) 
Carbonic  Oxide  (CO) 
Carbonic  Acid  (C02) 
Chlorine  (Cl) 
Cyanogen  (NC2) 
Marsh  Gas  (CH4) 
Olefiant  Gas  (C2H4) 
Ammonia  (NH3) 

Relative 
Densities. 

1 
1-10563 
•06926 
•97135 
•9545 
1-52907 
2-4222 
1-8019 
•562 
•982 
•5952 

Relative 
Specific 
Volumes. 

1 
•90446 
14-4383 
1-02945 
1-0476 
•6540 
•4128 
•5550 
1-779 
1-018 
T6801 

Mass  of  a 
Litre  in 
Grammes. 

1-2932 
1-4298 
•08957 
1-25615 
1-2344 
1-9774 
3-1328 
2-3302 
•727 
1-270 
•7697 

Volume  of 
a  Gramme 
in  Litres. 

•7733 

•6994 
11-16445 
•7961 
•8101 
•5057 
•3192 
•4291 
1-375 
•787 
1-2992 

Volume  of 
a  Pound  in 
Cubic  Feet. 

12-39 
11-20 

178-85 
12-75 
12-97 

8-10 
5-11 
6-87 
22-04 
12-61 
20-81 

By  means  of  these  data,  we  have  deduced  the  values 

°f  2^  '  where  v° is  the  volume  in  cubic  centimetres  of  one 

gramme  of  gas  at  0°  C.  and  a  pressure  of  one  atmo,  and 

l)pt  is  the  number  of  dynes  per  square  centimetre  in  a 

1  Everett's  '  Units  and  Physical  Constants.' 


ON  PERFECT  GASES.  89 

pressure  of  one  atmo,  (2)  p0  is  the  number  of  gramme- 
weights  (at  Paris)  per  square  centimetre  in  a  pressure  of 
one  atmo. 

(1)  (2) 

Air  2,871,000  2927 

Oxygen  2,596,000  2647 

Hydrogen  41,448,000  42256 

Nitrogen  2,955,000  3013 

Carbonic  Oxide  3,007,000  3066 

Carbonic  Acid  1,877,000  1914 

Chlorine  1,185,000  1208 

Cyanogen  1,593,000  1624 

Marsh  Gas  5,105,000  5205 

Olefiant  Gas  2,922,000  2980 

Ammonia  4,820,000  4917 

40.  As  an  illustration  of  the  first  law  of  thermo- 
dynamics in  its  application  to  perfect  gases,  let  us 
suppose  a  cylinder  fitted  with  an  air-tight  piston  to 
contain  a  mass  ra  of  any  kind  of  gas  at  temperature  ©, 
as  measured  by  the  air  thermometer,  and  let  the  volume 
be  altered  so  slowly  that  at  every  instant  the  gas  is  in 
a  state  of  equilibrium  and  its  temperature  kept  equal 
to  ©  by  the  conduction  of  heat  through  the  sides  of 
the  cylinder.  Then  the  force  exerted  by  the  gas  on  the 
piston  will  be  uniform  and  its  value  p,  per  unit  of  area, 
when  the  volume  is  v,  will  be  given  by  the  formula 

pv  =  inR®, 

so  that  the  state  of  the  gas  will  be  represented  on  an 
indicator  diagram  by  a  rectangular  hyperbola. 
The  work,  pdv,  done  by  the  gas  on  the  piston  during  a 
small  increase  of  volume  may  therefore  be  written 


90  ELEMENTARY    THERMODYNAMICS. 

Hence  when  the  volume  changes  from  ^  to  va,  the  work 
done  by  the  gas  will  be 


From  this  it  is  clear  that  when  the  volumes  form  a 
geometrical  progression,  the  corresponding  quantities  of 
work  will  form  an  arithmetical  progression. 

Again,  if  p1  be  the  initial  pressure,  the  work  done  by 
the  gas  becomes 


Hence  if  two  cylinders  of  the  same  volume  contain  any 
kinds  of  perfect  gas  at  different  temperatures  and  in  such 
quantities  that  the  pressure  is  the  same  in  both,  each  gas 
will  do  the  same  amount  of  work  in  expanding  by  the 
same  amount  at  constant  temperature. 

41.  A  much  more  important  example  is  obtained 
by  supposing  the  cylinder  to  contain  one  gramme  of 
gas,  the  temperature  of  which  is  slowly  raised  one 
degree  centigrade  (by  the  air  or  mercury  thermometer) 
by  imparting  heat  to  it  under  two  different  conditions  :  — 

(1)  Let  the  temperature  be  slowly  raised  while  the 
volume  is  kept  constant. 

The  increase  of  energy  in  this  case  is  equal  to  the 
'  specific  heat  of  the  gas  at  constant  volume,'  and,  as 
we  have  already  seen,  is  a  function  of  the  temperature 
alone,  or  a  constant.  It  is  usually  denoted  by  Cv  (in 
ergs). 

(2)  Let  the  temperature  and  volume  of  the  gas  be 
slowly  increased  in  such  a  way  that  the  pressure  remains 
constant. 


ON   PERFECT  GASES.  91 

The  heat  absorbed  by  the  gas  during  the  process  is  the 
'  specific  heat  of  the  gas  at  constant  pressure  '  and  is 
generally  written  Gp  :  the  increase  of  energy  is  the  same 
as  before,  since  the  energy  of  a  perfect  gas  in  a  state  of 
equilibrium  depends  only  on  the  temperature.  Hence  if 
W  be  the  work  done  by  the  gas,  we  have 


But  if  p  be  the  constant  pressure,  v  and  v'  the  initial 
and  final  volumes,  we  have  W  =  p  (v'  —  v),  which  is  equal 
to  the  constant  R,  by  the  relation  pv  =  lid.  Thus 


or  CP-CV  =  R  ...................  (16). 

This  simple  relation,  in  the  hands  of  Clausius  and 
Rankine,  furnished  some  of  the  earliest  triumphs  of  the 
mechanical  theory  of  heat. 

Prior  to  1850,  it  was  supposed  to  have  been  es- 
tablished by  experiment  that  the  specific  heat  of  a 
perfect  gas  depended  on  the  pressure,  but  in  that  year 
Clausius  was  led  by  the  new  theory  of  heat  to  assert 
that  the  specific  heat,  whether  at  constant  pressure 
or  at  constant  volume,  could  depend  only  on  the  tem- 
perature, and  he  conjectured  that  it  would  be  found 
to  be  constant.  The  fact  that  the  new  theory  dis- 
agreed with  the  results  of  accepted  experiments  led  to  an 
attack  upon  it  by  Holtzmann.  Three  years  later,  however, 
Regnault  published  his  splendid  experiments  on  the  specific 
heats  of  gases,  by  which  the  conclusions  drawn  from  the 
mechanical  theory  were  decisively  shown  to  be  true  ; 
the  specific  heat  of  a  permanent  gas  being  found  to  be 
independent  both  of  the  pressure  and  the  temperature. 


92  ELEMENTARY  THERMODYNAMICS. 

A  still  more  important  application  of  equation  (16) 
was  made  by  Rankine  a  little  later  in  the  same  year 
1850,  as  follows  : 

The  velocity  of  sound  in  air  depends  on  the  relation 
between  its  pressure  and  density  during  the  rapid  con- 
densations and  rarefactions  as  the  sound  travels  along. 
These  changes  of  pressure  and  density  occur  at  least 
hundreds  of  times  in  a  second,  and  consequently  the  heat 
developed  by  compression  has  not  time  to  get  away  by 
conduction  before  the  air  is  again  in  its  natural  state 
as  to  temperature  and  density.  A  formula  is  thence 
obtained  for  the  velocity  of  sound  in  air  which  contains 

C1 

the  ratio  of  the  two  specific  heats,  ^  ,  usually  written  k. 

L>v 

Comparing  the  formula  with  the  velocity  of  sound  in  air, 
as  observed  by  Bravais  and  Martens,  it  is  found  that 


Hence,  by  equation  (16),  we  have 


1-41  x  2,871,000 


=  9,873,000, 

and  Cv  =  7,002,000. 

If  CP,  cv,  be  the  specific  heats  in  calories,  we  have 

_    9,871,000 
p    41,539,759-8"       77' 
and  Co  =  -1686. 

The  value  previously  accepted  for  cp  at  atmospheric 


ON   PERFECT  GASES.  93 

pressure  was  '2669.     Regnault's  experimental  result,  ob- 
tained a  few  years  later,  was  cp  =  '2375. 

42.  There  is  another  method  of  estimating  the 
specific  heats  of  gases,  first  calculated  by  Clausius, 
which  is  often  useful  ;  viz.,  the  ratio  of  the  heat  re- 
quired to  raise  the  temperature  of  a  quantity  of  gas 
at  constant  pressure  or  volume  to  the  heat  required, 
under  the  same  conditions  as  to  pressure  or  volume,  to 
raise  the  temperature  of  an  equal  volume  of  air  to  the 
same  extent.  These  two  specific  heats  will  be  denoted 
by  yp  and  yv,  respectively. 

Since  we  have 

Cp  —  Cv  =  R, 
we  obtain,  for  air, 

9097 


*»     42350 
If  v'  be  the  volume  of  one  gramme  of  another  gas  at 
the  same  temperature  6  and  pressure  p  as  air,  and  if  R  be 
the  corresponding  value  of  R,  we  have 


pv  = 
Hence  R'  =  R  -  =  Rx,  say, 

where  x  is  the  relative  specific  volume  of  the  gas  com 
pared  with  air,  as  given  in  the  table. 
We  have,  therefore,  for  any  gas 


................  (17), 

a  formula  which  enables  us  easily  to  calculate  c/  from 
Regnault's  experimental  determination  of  cp'. 

Again,  the   quantity   of  heat   which  unit  volume  of 


94          ELEMENTARY  THERMODYNAMICS. 

the  gas  absorbs  when  its  temperature  is  slowly  raised  at 

C  ' 
constant  pressure  by  1°  C.  is    p,  .     Hence 


where  y  is  the  density  of  the  gas  compared  with  air,  as 
in  the  table. 

Substituting   for   cp   its   value  '2375,  as  found   by  Reg- 
nault,  we  get 


In  like  manner,  we  have, 


=  (7/  _  -2909)  &  (19). 

7cy 

By  means  of  these  formula?,  the  accompanying  table 
is  calculated  from  Regnault's  experimental  determinations 
of  the  specific  heats  at  constant  pressure  in  calories. 


Specific  heat  at 

constant  pressure. 

Compared 

In  calories. 

with  an  equal 
volume  of  air. 

Air                                     -2375 

1 

Oxygen  (0) 

•21751 

1-012 

Hydrogen  (H) 

3-40900 

•994 

Nitrogen  (N) 

•24380 

•997 

Carbonic  Oxide  (CO) 

•2450 

•985 

Carbonic  Acid  (C02) 

•2169 

1-396 

Chlorine  (Cl) 

•12099 

1-234 

Marsh  Gas  (CH4) 

•5929 

1-403 

Olefiant  Gas  (C2H4) 

•4040 

1-670 

Ammonia  (NH3) 

•5084 

1-274 

Specific  heat  at 
constant  volume. 

In  calories. 

Compared 
with  an  equal 
volume  of  air. 

k 

•1684 

1 

1-410 

•15501 

1-018 

1-403 

2-4114 

•992 

1-414 

•17266 

•996 

1-412 

•1728 

•978 

1-418 

•1717 

1-559 

1-263 

•0925 

1-330 

1-308 

•4700 

1-568 

1-260 

•3337 

1-946 

1-211 

•3923 

1-386 

1-296 

ON   PERFECT   GASES.  95 

43.  If  we  consider  a  gramme  of  perfect  gas  in 
a  state  of  equilibrium,  the  three  quantities  (p,  v,  6} 
satisfy  the  relation  pv  =  R9  and  therefore  any  two 
of  them  may  be  chosen  as  independent  variables.  A 
confusion  then  arises  as  to  the  meaning  of  a  partial 

differential    coefficient.      Thus    -^    will    have   different 

meanings  according  as  6  and  v  or  0  and  p  are  the 
independent  variables.  To  remove  this  difficulty,  Clau- 
sius  has  introduced  a  very  convenient  method  of  no- 
tation which  has  been  generally  adopted.  He  denotes 
the  partial  differential  coefficient  of  U  with  respect 

to  0  when  6  and  v  are  independent  variables  by  —  -  . 

Similarly,  the  partial  differential  coefficient  of  U  with 
respect  to  6  when  6  and  p  are  the  independent  variables 

dpU 

is  written  -%•=-  . 
au 

With  this  notation,  we  have,  by  Taylor's  theorem, 

JJT     dvU,a     deU  , 
dU=  -~  dd  +    ,     dv, 
do  dv 

or,  since  -t—  =  0  for  perfect  gases,  and  -"^  =  Cv, 


and  therefore  U=CV6  +  C'  ...................  (20), 

where    G'  is   an   arbitrary   constant   depending    on    the 
choice  of  a  standard  state. 

If  the  volume  and  temperature  of  the  gas  alter  so 
slowly  that  at  every  instant  the  gas  is  in  a  state  of 
equilibrium,  the  force  by  which  the  gas  is  compressed 
will  only  be  just  sufficient  to  overcome  the  pressure  of 
the  gas.  Hence 

dW=-pdv, 


96  ELEMENTARY   THERMODYNAMICS. 

and  therefore  the  equation 

dU=dW  +dQ 

becomes  d  U  =  dQ  —  pdv, 

whence  dQ  =  d  U  +  pdv 

=  CvdO+pdv 

=  Cvde  +  ™dv    ...............  (21). 

If  6  and  p  be  taken  for  independent  variables, 


or  thus  :  — 


=  Cvdd  +  {d(pv)-vdp} 

=  (Cv  +  R)d0-vdp 

=  CjjAO  -vdp  ........................  (22)  ; 


73  f\ 

--  dp,  since  pv  =  Ed, 


. 
p    • 

If  we  wish  to  take  p  and  v  as  the  independent 
variables,  we  may  write  ^  (pdv  +  vdp)  for  dO  in  (21): 
hence 


V  +  v 

—  -g  —  pdv  +  ~  vdp 

C  C 


Cv 
p 

(23). 


ON   PERFECT  GASES.  97 

44.  The  gas  will  generally  be  losing  or  gaining  heat 
by  conduction  through  the  sides  of  the  cylinder,  but  if  a 
good  non-conducting  material  be  used  either  in  making 
the  cylinder  or  as  a  covering  for  it,  the  rate  at  which  heat 
is  lost  or  gained  in  this  way  will  become  very  small.  We 
are  thus  led  to  imagine  an  ideal  case  in  which  no  heat  at 
all  can  enter  or  leave  the  gas  during  the  expansion.  We 
have  then  dQ  =  0,  and  therefore,  by  equation  (21), 


Integrating, 

Cv  log  6  +  E  log  v  =  a  constant. 

0 

Writing       Cp  -  Cv  for  R,  and  k  for  ^  ,  we  get 

L/v 

log  0  +  (k  —  1)  log  v  =  a  constant, 
or  dtf~l  =  a  constant. 

Hence  if  00,  Vo  are  the  values  of  6,  v  at  any  instant,  we 
shall  have 


n 


Suppose,  for  example,  that  a  quantity  of  air,  originally 
at  the  freezing  point,  is  contained  in  a  cylinder  imperme- 
able to  heat,  and  let  the  piston  be  slowly  pushed  in  till  the 
volume  is  reduced  by  half:  then  00=  273  and  k  =  1'410, 
whence 

^  =  2-  =  1-329, 

and    0  =  273  x  1'329  =  363,  or  ff  =  363  -  273  =  90. 

The  compression  therefore  causes  the  temperature  to  rise 

90°  C. 

p.  7 


ELEMENTARY   THERMODYNAMICS. 


Again,  if  v  =  -£  ,  and  00  =  273,  as  before,  we  have 


and  0  =  273  x  1-765  =  482,  or  ff  =  482  -  273  =  209. 

To  find  the  relation  between  6  and  p,  we  may  either 
substitute  for  v  in  equation  (24)  by  means  of  the  relation 
pv  =  R6  :  thus 

0 


or  we  may  work  from  equation  (22)  :  thus 
Cpdd-Redp  =  Q, 

whence       Cp  log  0-(Cp-  Cv)  logp  =  a  constant, 
or  k  log  6  -  (k  -  1  )  log  p  =  a  constant, 

Qk 

or  ^=1  =  a  constant. 

Hence,  just  as  before, 


The  relation  between  p  and  v  is 

Cvvdp  +  Cppdv  =  0, 

that  is,  C,3P  +  C^  =  0. 

p        p  v 

Hence  I0g^  +  k  log  v  =  constant. 

If  p0,   v.  are  corresponding  values  of  p  and  v,  this 
becomes 


ON   PERFECT   GASES. 

L* 


99 


.(26). 


P 

Po 

The  work  done  by  the  gas  in  altering  its  volume  from 
v1  to  v.,  is 

P'  j          *  P2  rfy 

I    jrav  =  £>0^o   I     -T 
JW,*  /    1 

~&=i  ft*1 

When  any  substance  is  slowly  compressed  or  rarefied 
in  a  cylinder  impermeable  to  heat,  the  corresponding 
indicator  diagram  is  called  by  Rankine  an  Adiabatic 
curve,  and  by  Prof.  Gibbs,  an  Isentropic  curve,  because 
the  Entropy,  a  quantity  which  will  be  discussed  in  the 
next  chapter,  remains  constant  throughout  the  operation. 
On  the  other  hand,  when  the  temperature  is  kept  constant 
during  the  process,  the  corresponding  diagram  is  known 
as  an  Isothermal  curve. 

The  equation  of  an  adiabatic  curve  for  a  perfect 
gas  being 


Isothermal. 


Adiabatic. 


pvfi  =  constant, 
and  that  of  an  isothermal 

pv  =  constant, 


7—2 


100 


ELEMENTARY   THERMODYNAMICS. 


it  is  clear,  since  k  is  >  1,  that  if  an  •  isothermal  and  an 
adiabatic  curve  cross  one  another  at  any  point,  the  adia- 
batic  curve  will  be  more  steeply  inclined  to  the  axis  of  v 
at  the  point  of  intersection  than  the  isothermal. 

45.  We  are  now  in  a  position  to  consider  a  very 
important  example  of  the  conversion  of  heat  into  me- 
chanical work,  which  will  prepare  the  way  for  the  next 
chapter. 

Let  us  suppose  a  gramme  of  any  perfect  gas  contained 
in  a  cylinder  fitted  with  an  air-tight  piston,  as  shown  in 
the  sketch,  the  piston  and  every  part  of  the  cylinder 
except  the  bottom  being  impermeable  to  heat,  while  the 
bottom  is  supposed  to  have  so  small  a  capacity  for  heat 


that  the  heat  required  to  raise  its  temperature  may  be 
neglected.  Such  a  cylinder  cannot  be  constructed  in 
practice.  It  is  merely  a  limit  towards  which  we  may 
approximate  very  closely,  but  which  can  never  be  actually 
attained. 

Let  us  also  suppose  that  there  are  two  bodies  A,  £  of 
any  kind,  the  temperatures  of  which,  as  shown  on  the 
air  thermometer,  are  kept  constantly  equal  to  00,  and  6bi 


ON   PERFECT  GASES. 


101 


respectively  ;    B  being  colder  than  A,  and  of  the  same 
temperature  as  the  gas  inside  the  cylinder.     Also  let  G  be 


a  perfect  non-conductor  on  which  the  cylinder  can  be  set 
for  any  length  of  time  without  losing  or  gaining  heat. 

Then-  let  the  gas  inside  the  cylinder  be  made  to 
undergo  the  following  complete  cycle  of  operations,  the 
conception  of  which  is  due  to  Carnot. 

(1)     Let  the  cylinder  be  placed  on  (7,  and  then  press 


the  piston  down  very  slowly  until  the  temperature  of  the 
gas  rises  from  6b  to  0a,  the  operation  being  represented  on 
the  indicator  diagram  by  the  adiabatic  curve  PQ. 

(2)     Transfer  the  cylinder  to  A  and  then  let  the  piston 


102  ELEMENTARY   THERMODYNAMICS. 

be  drawn  out  so  slowly  that  the  temperature  of  the  gas  is 
prevented  from  falling  by  the  passage  of  heat  from  A 
through  the  bottom  of  the  cylinder.  Let  the  heat  ab- 
sorbed by  the  gas  be  called  qa,  and  suppose  the  operation 
represented  on  the  diagram  by  the  isothermal  curve  QR. 

(3)  Place  the  cylinder  again  on  C  and  then  slowly 
draw  the  piston  further  out  until  the  temperature  of  the 
gas  again  becomes  Ob.     The  corresponding  curve  in  the 
figure  is  the  adiabatic  US. 

(4)  Lastly,  let  the  cylinder  be  placed  on  B,  and  then 
by  forcing  in  the  piston  slowly  let  the  volume  be  dimin- 
ished at  the  constant  temperature  6b  until  the  initial  state 
of  the  gas  is  attained,  the  heat  given  to  B  being  denoted 
by#, 

The  heat,  qa,  which  is  absorbed  by  the  gas  in  the  second 
operation  is  the  thermal  equivalent  of  the  work  done  by 
the  gas  in  that  operation,  since  the  temperature  is  con- 
stant. Hence  if  vq,  VK  be  the  volumes  of  the  gas  cor- 
responding to  the  points  Q,  E,  we  have 


Similarly  qb  =  R0b  iog  ^  . 

vp 

Now  since  the  equation  to  an  adiabatic  curve  is 

Qvk~l  =  a  constant, 
we  get,  from  the  adiabatic  curve  PQ, 


ON   PERFECT   GASES.  103 


and  from  the  adiabatic  curve  US 

_  /v^k-1 

eb 

Hence  — 


and  therefore  TT  =  TT     •  ...(27). 

0ffl        t/6 

The  total  work  done  by  the  gas  during  the  cycle  will 
be  qa  —  qb,  by  the  first  law  of  thermodynamics,  and  since 
the  process  may  be  repeated  indefinitely,  we  may  consider 
that  this  work  has  been  transformed  out  of  heat. 

The  ratio  of  the  work  obtained  from  the  cycle  to  the 
mechanical  equivalent  of  the  heat  absorbed  from  the  hotter 

body,  is  —  —  —  ,  or  1  —  -£  ,  and  is  therefore  the  same  what- 

ffa  "a 

ever  kind  of  perfect  gas  be  operated  upon.     It  is  called 
the  '  efficiency  '  of  the  cycle. 


CHAPTER  III. 
CARNOT'S  PRINCIPLE. 

46.  ACCORDING  to  the  principle  of  the  Conservation  of 
Energy,  the  total  energy  of  any  material  system  which  is 
prevented  from  exchanging  energy  with  external  systems 
and  from  radiating  energy  into  infinite  space,  remains 
invariable.  But  though  the  total  energy  continues  the  same 
in  amount,  it  may  assume  different  forms,  energy  being 
convertible,  under  the  proper  circumstances,  out  of  any 
one  form  into  an  equivalent  in  any  other  form. 

The  foundations  of  the  important  and  interesting 
subject  of  the  Transformation  of  Energy  were  laid  by 
Carnot  in  a  profound  essay  published  in  1824.  This 
remarkable  and  curious  work  was,  unfortunately,  vitiated 
by  the  false  view  then  prevalent  as  to  the  nature  of  heat. 
It  is  satisfactory,  however,  to  find  that  before  he  died  in 
1832,  Camot  had  not  merely  emancipated  himself  from 
the  doctrine  of  caloric1,  but  had  made  a  good  approxima- 
tion to  the  mechanical  value  of  a  calorie2. 

Although  Carnot  is  now  admitted  to  have  been  one  of 

1  See  Bertrand's  Thermodynamics. 

2  Carnot  concluded  that  a  calorie  was  equivalent  to  37000  gramme- 
centimetres.     This  is  a  more  accurate  estimate   than  Mayers  (36500) 
obtained  in  1842. 

See  Carnot's  '  Reflexions  sur  la  puissance  motrice  du  feu.'  (Gauthier- 
Vfflars,  Paris,  1878.) 


CARNOT'S  PRINCIPLE.  105 

the  greatest  men  produced  by  France,  and  his  principle  to 
be  one  of  the  greatest  discoveries  yet  made  in  science,  his 
work  failed  to  attract  attention  for  some  time  after  it  was 
published.  But  ten  years  later,  it  was  brought  prominently 
forward  by  Clapeyron  who  exhibited  it  in  a  more  elegant 
form  by  means  of  Watt's  indicator  diagram.  Still  very 
little  progress  appears  to  have  been  made  with  the  subject 
until  the  appearance  of  the  present  theory  of  heat. 

In  1850,  soon  after  the  establishment  of  the  mechani- 
cal theory  of  heat,  it  was  shown  by  Clausius  that  Carnot's 
reasoning  could  easily  be  modified  so  as  to  be  consistent 
with  the  new  theory  and  he  then  proceeded  to  recast  it 
in  its  present  form.  In  this  undertaking  he  was  ably 
seconded  about  the  same  time  by  Rankine,  and  a  little 
later  by  Sir  W.  Thomson,  to  whom  we  owe  a  truly 
'  absolute'  scale  of  temperature. 

47.  Before  we  can  make  use  of  Carnot's  principle, 
it  is  necessary  to  introduce  the  conception  of  reversible 
operations.  A  few  illustrations  will  make  this  idea  clear. 

(1)  When  two  pieces  of  metal  C,  D  are  in  contact  at 
the  same  temperature  without  being  rubbed  together,  no 
heat  will  pass  from  one  to  the  other.  But  if  the  tempe- 
rature of  G  be  raised  ever  so  little,  heat  will  at  once  begin 
to  flow  across  the  junction,  and  if  the  small  difference  of 
temperature  be  maintained  by  any  means  for  a  sufficient 
time,  any  amount  of  heat  may  be  transferred  from  one 
body  to  the  other.  In  like  manner,  by  slightly  raising 
the  temperature  of  D  without  altering  that  of  C,  a  finite 
quantity  of  heat  may  be  made  to  pass  in  the  opposite 
direction.  The  reversible  limit  towards  which  we  may 
approximate  as  near  as  we  please  without  ever  being  able 


106  ELEMENTARY   THERMODYNAMICS. 

actually  to  reach,  is  that  of  causing  heat  to  flow  in  either 
direction  without  any  difference  of  temperature  at  all. 

(2)  Let  us  suppose  that  a  cylindrical  rod,  which  is 
placed  parallel  to  the  axis  of  x,  is  made  to  move  with  an 
acceleration  to  the  right.  Then  if  P,  Q  be  any  two 


sections  of  the  rod,  it  is  clear  that  the  forces  exerted  on 
the  portion  PQ  across  the  sections  P  and  Q  will  not 
generally  be  equal.  For  example,  if  these  forces  be  ten- 
sions and  no  other  forces  act  parallel  to  the  axis  of  x,  the 
tension  across  the  section  Q  must  be  greater  than  the 
tension  across  the  section  P.  If,  on  the  contrary,  the  rod 
moves  with  an  acceleration  to  the  left,  the  tension  across 
the  section  Q  will  be  less  than  that  across  P.  The  in- 
ternal state  of  the  rod  is  therefore  different  in  these  two 
opposite  motions;  but  we  may  make  the  difference  as 
small  as  we  please  without  actually  causing  the  accelera- 
tions to  become  zero.  Then  by  allowing  sufficient  time, 
any  finite  change  of  velocity  may  be  produced.  The  limit 
in  this  case  which  cannot  be  exactly  attained,  is  that  of 
causing  the  velocity  parallel  to  the  axis  of  x,  of  the 
portion  PQ,  to  alter  without  any  difference  in  the  ten- 
sions across  the  sections  P,  Q.  If,  in  this  limiting  case, 
there  be  no  electrification  and  heat  be  imparted  to  or 
abstracted  from  the  rod  in  a  reversible  manner,  the 
operation  which  the  rod  undergoes  will  be  completely 
reversible. 

(3)     If  a  quantity  of  gas  be  contained  in  a  cylinder 
fitted  with  an  air-tight  piston,  it  is  evident  that  the  piston 


CARNOT'S  PRINCIPLE.  107 

cannot  be  pushed  in  without  exerting  a  greater  force  on 
the  gas  than  when  it  is  at  rest :  similarly  if  the  expansion 
of  the  gas  force  the  piston  out  a  little,  the  pressure  of  the 
gas  on  the  piston  will  be  less  than  if  the  piston  had  not 
moved.  But  in  both  cases,  by  making  the  velocity  of  the 
piston  slow  enough,  we  may  approximate  to  an  ideal 
operation  in  which  the  pressure  of  the  gas  on  the  piston 
is  the  same  as  if  it  was  at  rest.  If,  in  this  ideal  case,  the 
temperature  of  the  gas  be  kept  constant,  or  caused  to 
depend  only  on  the  volume,  it  is  clear  that  the  gas  may 
be  compressed  back  again  by  exactly  the  same  force  which 
it  has  overcome  in  expanding. 

(4)  A  remarkable  example  of  reversibility  may  occur 
when  a  cylinder  of  moderate  size  is  filled  with  water  and 
steam  instead  of  gas.     If,  as  before,  the  piston  be  pushed 
in,  equilibrium  will  be   impossible  within   the   cylinder, 
both  as  regards  pressure  and  temperature,  so  long  as  the 
motion   lasts.     But    if,   after   the   motion   of  the   piston 
ceases,  the  temperature  be  reduced  to  the  same  value  as 
before  and  there  be  still  room  for  steam  to  be  formed,  it  is 
found  by  experiment  that  the  pressure  is  unaltered  by  the 
compression.     We  are  therefore  led  to  imagine  a  limiting 
reversible  case  in  which  steam  is  converted  into  water  or 
water  into  steam  without  any  change  either  in  the  pres- 
sure or  the  temperature.     The  indicator  diagram  repre- 
senting such  a  process  will  be  a  straight  line  parallel  to 
the  axis  of  v. 

(5)  Again,  let  us  suppose  the  cylinder  to  contain  a 
saturated  aqueous  solution  of  a  salt  which  deposits  the 
anhydrous  salt  and  not  a  hydrate ;  for  example,  Nitrate  of 
Lead.  Also  let  there  be  a  quantity  of  the  salt  undissolved. 
Then  if  the  temperature  be  slowly  changed,  the  solution 


108  ELEMENTARY   THERMODYNAMICS. 

will  alter  in  strength  by  diffusion  and  it  is  clear  that  the 
operation  will  ultimately  be  reversible. 

In  every  case  of  reversibility,  the  process  which  the 
system  undergoes  is  a  non-frictional  process ;  but  it  does 
not  follow,  conversely,  that  every  non-frictional  process  is 
reversible.  We  have  already  had  examples  of  irreversible 
non-frictional  processes  in  Article  25,  Chap.  I.  Again,  if 
a  cylinder  contain  a  saturated  solution  of  Sulphate  of 
Magnesium  together  with  an  excess  of  the  salt,  increase 
of  temperature  will  increase  the  strength  of  the  solution 
by  causing  some  of  the  salt  to  be  dissolved ;  but  on  again 
reducing  the  temperature,  the  solution  will  deposit,  not  the 
anhydrous  salt,  but  a  hydrate.  The  process  is  therefore 
not  reversible,  even  though  it  is  a  non-frictional  process. 

48.  Carnot's  principle  relates  to  the  transformation 
of  heat  into  mechanical  work. 

Suppose  that  any  material  system  whatever,  which 
may  be  electrified  in  any  way  we  please,  is  made  to 
undergo  a  change  of  state  during  which  there  are  no 
external  electric  influences,  so  that  the  energy  cannot  be 
affected  by  external  systems  except  by  the  conduction  or 
radiation  of  heat  and  by  the  doing  of  mechanical  work. 
Then,  unless  the  change  of  energy  is  zero,  we  cannot 
assert  that  the  mechanical  work  obtained  from  the  system 
has  been  transformed  out  of  the  heat  absorbed  by  it,  for 
part  of  the  work  may  owe  its  origin  to  the  change  in  the 
energy  of  the  system.  It  will  therefore  be  necessary  for 
us,  in  the  first  instance,  to  restrict  ourselves  to  the  con- 
sideration of  complete  cyclical  processes. 

Again,  if  the  system  be  gaming  or  losing  heat  at  any 
finite  speed  in  any  part  A,  either  by  conduction  or  radia- 


CARNOT'S  PRINCIPLE.  109 

tion,  it  will  be  impossible  to  keep  the  temperature  of  A 
strictly  uniform  and  constant ;  but  by  taking  sufficient 
precautions,  we  shall  sometimes  be  able  to  prevent  the 
temperature  of  every  part  of  A  from  differing  by  more 
than  an  infinitesimal  amount  from  a  given  constant  tem- 
perature. For  example,  if  A  be  a  thin  closed  metallic 
vessel  filled  with  a  very  volatile  liquid  and  its  vapour, 
any  difference  in  the  temperatures  of  different  parts  of 
A  would  cause  an  explosive  formation  or  condensation 
of  vapour,  by  which  the  temperature  of  A  would  quickly 
be  rendered  uniform.  Also  if  A  consist  of  two  parts  of 
which  the  upper  part  is  a  cylinder  fitted  with  an  air-tight 
piston  and  connected  by  a  pipe  with  the  lower  part,  it 
will  be  practically  possible,  by  moving  the  piston  in  or 
out,  to  keep  the  temperature  of  A  not  merely  uniform, 
but  constant. 

Let  us  now  suppose  the  system  to  undergo  any  cyclical 
process  during  \vhich  it  is  prevented  from  receiving  or 
losing  heat,  either  by  conduction  or  radiation,  except  in 
two  parts  A,  B.  Then  as  we  approach  the  ideal  case  in 
which  the  temperatures  of  A  and  B  are  kept  uniform  and 
constant,  we  assume  as  an  axiom  that,  whatever  the  two 
parts  A,  B  maybe,  no  mechanical  work  can  be  obtained 
from  the  system  during  the  cycle  so  long  as  the  tempera- 
tures of  A  and  B  are  equal.  If  work  is  gained  from  the 
cycle,  the  temperatures  of  A  and  B  must  therefore  be 
different. 

This  important  fundamental  axiom  is  substantially 
due  to  Carnot  and  its  great  merit  arises  from  the  fact  that 
the  operations  referred  to  constitute  a  complete  cycle.  It 
is  not  generally  true  when  the  cycle  is  incomplete,  as  will 
appear  later  on.  The  preceding  is  probably  the  simplest 


110  ELEMENTARY  THERMODYNAMICS. 

way  in  which  it  can  be  stated ;  but  in  this  book  we  shall 
follow  the  usage  of  all  existing  text-books  by  employing  it 
in  a  form  differing  but  little  from  that  in  which  it  was 
originally  stated  by  Carnot  himself.  We  shall  suppose 
that  the  system  undergoes  a  complete  cycle  such  that  all 
the  heat  absorbed,  whether  by  conduction  or  radiation, 
comes  from  two  bodies  A,  B,  whose  temperatures  are  kept 
uniform  and  constant,  and  that  conversely,  all  the  heat 
given  out  goes  to  the  same  two  bodies  A,  B ;  and  we  shall 
assume  as  an  axiom  that  no  mechanical  work  can  be 
obtained  from  the  system  during  the  cycle  so  long  as  the 
temperatures  of' A  and  B  are  equal. 

If  the  given  system  can  lose  or  gain  heat  by  conduc- 
tion only,  it  will  be  clearly  possible  to  keep  the  tempera- 
tures of  A  and  B  practically  uniform  and  constant  by 
causing  all  the  heat  exchanged  between  the  system  and  A 
and  B  to  be  conducted  respectively  through  two  immense 
intermediate  bodies,  or  systems  of  bodies,  A',  B'.  The 
condition  that  the  whole  of  the  heat  lost  from  the  given 
system  should  be  transmitted  by  A'  and  B'  to  A  and  B, 
and  conversely,  can  easily  be  satisfied.  We  take  A'  and 
B'  unelectrified,  and  consequently  free  from  all  external 
electric  influences ;.  and  then  make  them  both  undergo 
complete  cyclical  processes  during  the  same  time  as  the 
given  system,  without  either  doing  work  upon  them,  or 
obtaining  work  from  them.  In  these  cycles,  the  total 
amount  of  heat  absorbed  by  either  A'  or  B'  is  obviously 
zero.  Hence  if  we  suppose  that  both  A'  and  B'  are  pre- 
vented from  radiating  energy  themselves,  and  from  re- 
ceiving the  radiation  of  other  bodies;  and  also  that  A' 
can  only  exchange  heat  by  conduction  with  the  given 
system  and  with  A,  B'  with  the  given  system  and  with  B; 


CARNOT'S  PRINCIPLE.  Ill 

it  follows  from  Art.  15,  that  if  we  neither  rub  the  given 
system  nor  A  nor  B  against  A'  or  B',  the  quantities  of 
heat  conducted  during  the  cycle  from  the  given  system 
into  A'  and  B'  will  be  respectively  equal  to  the  quantities 
of  heat  obtained  by  A  from  A',  and  by  B  frorn  B,  in  the 
same  time ;  and  conversely. 

If  the  given  system  can  lose  or  gain  heat  by  radiation 
only,  we  suppose  that  all  the  heat  is  exchanged  by  radia- 
tion, not  with  A  and  B  directly,  but  with  two  inter- 
mediate bodies  A',  B',  which  are  the  same  as  before  except 
that  they  now  exchange  heat  with  the  given  system  by 
radiation.  Also  if  we  suppose  that  no  radiation  is  trans- 
mitted across  the  spaces  A",  B",  which  separate  the  given 
system  from  A'  and  B',  except  the  radiation  passing 
between  the  system  and  A'  and  B',  it  is  clear  that  if  the 
spaces  A",  B"  be  in  exactly  the  same  condition  at  the  end 
of  the  cycle  as  at  the  beginning  of  it,  the  quantities  of 
heat  lost  by  the  given  system  to  A'  and  B  during  the 
cycle  will  be  respectively  equal  to  the  quantities  derived 
in  the  same  time  by  A  from  A'  and  by  B  from  B ;  and 
conversely.  Unless  this  condition  is  fulfilled,  the  system 
cannot  be  said  to  exchange  heat  with  the  two  bodies 
A,  B,  only.  For  example,  if  the  system  and  A',  B'  be  at 
great  distances  from  one  another,  and  be  prevented  from 
emitting  radiation  until  the  beginning  of  the  cycles  which 
they  undergo,  these  cycles  may  be  completed  before  the 
radiation  emitted  by  any  one  of  them  can  get  to  the 
other :  in  this  case,  the  system  would  be  considered  to 
undergo  a  cycle  in  which  the  whole  of  the  heat  given  out 
was  supposed  to  be  radiated  into  infinite  space. 

Since  length  of  time  is  the  essential  element  in  all 
non-frictional  processes,  with  the  sole  exception  of  the 


112  ELEMENTARY   THERMODYNAMICS. 

non-frictional  process  by  which  the  motion  of  the  centre 
of  mass  is  varied,  which,  however,  cannot  form  a  complete 
cycle  unless  the  process  is  indefinitely  slow,  it  is  evident 
that  if  the  cycle  undergone  by  the  given  system  be  re- 
versible, the  cycles  undergone  by  A'  and  B'  will  also  be 
reversible ;  and  consequently  the  system  will  then  absorb 
or  give  out  heat  at  the  temperatures  of  A  and  B  only. 

49.  The  quantities  of  heat  absorbed  by  the  material 
system  from  the  two  bodies  A,  B,  during  any  complete 
cycle  cannot  both  be  positive.  For  we  could  then,  by 
expending  work  in  friction,  cause  the  system  to  undergo 
a  cycle  of  operations  in  which  a  positive  quantity  of  heat 
was  absorbed  from  one  of  the  bodies  A,  B,  and  no  heat  at 
all  either  received  from  or  parted  with  to  the  other.  In 
other  words,  we  should  be  able  to  take  heat  from  a  body 
whose  temperature  was  uniform  and  constant,  and  trans- 
form it  into  work  without  the  presence  of  any  other  body 
of  different  temperature,  contrary  to  Carnot's  axiom. 

If  the  cycle  be  reversible,  the  quantities  of  heat  ab- 
sorbed from  A  and  B  cannot  both  be  negative ;  for  by 
simply  reversing  the  cycle  we  should  obtain  another  cycle 
in  which  the  quantities  of  heat  absorbed  were  both  positive. 
Hence,  in  any  reversible  cycle,  a  positive  quantity  of  heat 
is  taken  from  one  of  the  bodies  A,  B,  and  a  positive 
quantity  given  to  the  other.  Consequently,  if  qa  be  the 
heat  absorbed  from  A  in  a  reversible  cycle  and  qb  the 
heat  given  to  B,  qa  and  qb  will  be  of  the  same  sign  and 
their  ratio  positive.  If  the  cycle  be  irreversible,  qa  and  qb 
may  either  be  of  the  same  or  of  opposite  signs  and  their 
ratio  either  positive  or  negative. 

Again,  suppose  that  any  two  material  systems  what- 


CARNOT'S  PRINCIPLE.  113 

ever,  X,  X',  undergo  any  complete  cycles  of  operations 
during  which  they  can  only  exchange  heat  with  the  two 
bodies  A,  B,  and  let  qa,  qa'  be  respectively  the  quantities 
of  heat  absorbed  from  A,  and  qb,  qb  the  quantities  restored 
to  B.  Then  if  the  cycle  undergone  by  X  be  reversible. 

the  ratio  —.  cannot  be  greater  than  the  positive  ratio  — 
qi  qi 

when  qb  is  positive,  nor  less  than  —  when  qb  is  negative. 

For  if  the  cycle  undergone  by  X  be  reversed,  the  system 
X  will  then  absorb  qb  from  B  and  give  back  qa  to  A.  We 
may  therefore  by  increasing  or  decreasing  the  system  in 
the  proper  proportion,  cause  it  to  absorb  qb'  from  B  and 

give  up  qa  —  to  A.  Combining  this  new  cycle  with  that  of 
X',  we  get  a  new  cycle  Y,  in  which  no  heat  is  exchanged 
with  B  while  a  quantity  qa'  —  qa  is  absorbed  from  A  . 

Hence,  by  Carnot's  axiom,  qa'  —  qa        cannot  be  positive, 

and  therefore   —.  cannot  be  greater  than  —  when  qb   is 

qb  qb 

positive,  nor  less  when  qb  is  negative. 

If  the  cycle  undergone  by  X'  be  also  reversible,  the 
cycle  Y  will  be  reversible,  and,  by  simply  reversing  it,  we 

may  show  that  qa'  —  qa  —  cannot  be  negative.  Hence 
qa'  —  qa  =  0,  so  that,  when  both  cycles  are  reversible, 


(28), 


whatever  be  the  natures  of  the  systems  X,  X',  or  of  the 
reversible  cycles  which  they  undergo. 

p.  8 


114  ELEMENTARY  THERMODYNAMICS. 

If  the  system  X  undergo  a  reversible  cyclical  process 
during  which  it  can  only  exchange  heat  with  two  other 
bodies  C,  D,  whose  temperatures  are  kept  uniform  and 
constant,  it  is  easily  proved  that,  whatever  the  bodies  C, 
D  may  be,  if  the  temperature  of  G  be  equal  to  that  of  A 
and  the  temperature  of  D  equal  to  that  of  B,  we  shall 
have 


where  qe  is  the  heat  absorbed  from  C  and  q^  the  heat 
given  to  D. 

The  ratio  —  for  a  reversible  cycle  can  therefore  depend 

only  on  the  temperatures  of  A  and  B,  and  is  independent 
of:— 

(1)  The  natures  of  the  bodies  A  and  B. 

(2)  The  nature  of  the  system  which  undergoes  the 
reversible  operation. 

(3)  The  nature  of  the  reversible  operation  itself. 
50.     The   different   temperatures   of  bodies   may  be 

conveniently  distinguished  by  applying  a  different  number 
to  each  temperature.  These  numbers  are  chosen  by  Sir 
W.  Thomson  in  such  a  way  that  the  numbers  0a,  6b, 
corresponding  to  any  two  temperatures  A,  B,  satisfy  the 
relation 

2?  =  ?»  (9Q\ 

ea    eb~  ^' 

where  qa  is  the  heat  absorbed  from  A  and  qb  the  heat 
restored  to  B  in  any  complete  reversible  cycle  working 
between  A  and  B.  We  are  then  at  liberty  to  fix  the 
value  of  any  one  temperature  we  please,  or,  better  still, 
we  may  choose  the  numbers  so  that  the  temperatures  of 


CARNOT'S  PRINCIPLE.  115 

the  ordinary  freezing  and  boiling  points  differ  by  100. 
Such  a  scale  of  temperature  is  known  as  Thomson's  ab- 
solute scale,  because  it  is  independent  of  the  particular 
properties  of  any  substances  or  class  of  substances.  It 
obviously  coincides  very  nearly  with  the  scale  of  the  air 
thermometer,  or  with  that  of  the  centigrade  mercuiy 
thermometer  increased  by  273.  The  freezing  point  on 
Thomson's  absolute  scale  will  therefore  be  about  273  and 
the  boiling  point  373.  Also  since  qa  and  <?&  are  of  the  same 
sign,  it  is  evident  that  all  the  numbers  which  denote 
temperatures  on  Thomson's  scale  are  positive. 

It  should  be  noticed  that  the  numbers  which  dis- 
tinguish temperatures  might  have  been  chosen  in  many 
different  ways  so  as  to  be  consistent  with  Carnot's  prin- 
ciple, but  we  could  not  have  chosen  them  arbitrarily. 
Thus  we  could  not  have  taken  Ba  and  6b  so  that 

qa  =     Og 
•qb     I+Ob' 

for  then  we  should  also  have  had  —  =  , — V  and  —  =      "• ; 

qc      i  +  c/c         qc      1  +  c/c 

which  would  have  involved  a  contradiction. 

On  the  ordinary  thermometers,  the  different  tempera- 
tures are  distinguished  by  different  numbers,  the  choice 
of  which  depends  on  some  special  property  of  the  ther- 
mometric  substance  which  is  not  possessed  exactly  by  any 
other  substance.  Hence  if  two  thermometers  be  con- 
structed with  different  thermometric  substances  and  be 
graduated  in  the  usual  way,  so  that  they  show  the  same 
readings  at  the  freezing  and  boiling  points,  they  will  not 
generally  agree  at  other  temperatures. 

For  the  future,  when  we  speak  of  temperature,  we  shall 

8—2 


116  ELEMENTARY   THERMODYNAMICS. 

always  understand  Thomson's  absolute  scale,  unless  it  is 
specially  stated  otherwise. 

If  we  put  6b  =  0,  equation  (29)  becomes 

t=^' 

from  which  we  deduce  qb  =  0.  This  result  probably  means 
that  at  the  absolute  zero  of  temperature,  the  system 
possesses  no  non-mechanical  kinetic  energy.  For  ex- 
ample, ice  is  nearly  incompressible  and  its  specific  heat 
may  be  supposed  independent  of  the  pressure  and  tem- 
perature and  equal  to  '5  calorie.  Hence,  since  no  change 
of  aggregation  occurs  between  absolute  zero  and  the 
melting  point,  the  non-mechanical  kinetic  energy1  of  a 
gramme  of  ice  at  0°  C.  may  be  taken  to  be  about  136'5 
calories,  or  5,670  million  ergs,  or  415  foot-pounds.  In 
English  measure,  the  non-mechanical  kinetic  energy  of  a 
pound  of  ice  at  0°  C.  will  be  about  190,000  foot-pounds. 
But  if  every  particle  be  moving  with  the  same  non- 
mechanical  velocity  of  v  centimetres  per  second,  the 
non -mechanical  kinetic  energy  of  a  gramme  will  be  ^v2 
ergs:  hence 

t>3  =  ll,340x  106, 

or  v  =  106,500. 

Thus  the  non-mechanical  velocity  of  each  particle  is 
106,500  centimetres,  or  T065  kilometres,  or  1164  yards, 
per  second.  This  is  equal  to  a  velocity  of  63'9  kilometres, 
or  nearly  40  miles,  per  minute.  In  the  case  of  iron,  the 

1  When  a  solid  body  is  raised  in  temperature  without  being  melted, 
the  increase  of  energy  is  probably,  to  a  large  extent,  non-mechanical 
kinetic  energy,  and,  in  a  rough  calculation  like  the  present,  may  be 
taken  to  be  entirely  such. 


CARNOT'S  PRINCIPLE.  117 

specific  heat  is  about  ^th  of  a  calorie,  and  therefore  the 
non-mechanical  kinetic  energy  of  a  gramme  of  iron  at 
0°  C.  is  about  1,260  million  ergs.  Thus  the  non-me- 
chanical velocity  of  each  particle  is  about  50,000  centi- 
metres, or  '5  kilometres,  or  547  yards,  per  second ;  that 
is,  30  kilometres,  or  nearly  19  miles,  per  minute. 
The  non-mechanical  motions  are  too  minute  to  be  ob- 
served, even  with  the  most  powerful  microscope,  but  it 
appears  from  theory  that  the  maximum  non-mechanical 
displacement  of  a  particle  in  a  solid  is  probably  less 
than  the  200  millionth  of  a  centimetre  (500  millionth 
of  an  inch).  Hence  since  each  particle  of  a  solid  describes 
an  oval  of  some  sort,  the  non-mechanical  motions  in  iron 
at  0°  C.  must  be  reversed  in  any  given  direction  at  least 
1012  times  in  a  second. 

51.  In  a  complete  cycle  working  between  A  and  B, 
the  quantities  qa  and  qb  may  be  either  positive  or  negative ; 
but  since  a  positive  quantity  of  heat  cannot  be  taken  both 
from  A  and  B,  it  is  evident  that  heat  can  only  be  trans- 
formed into  work  during  the  cycle  when  a  positive  quantity 
of  heat  is  taken  from  one  of  the  two  bodies  A,  B,  and  a 
positive  quantity  given  to  the  other,  that  is,  when  qa  and 
qb  are  of  the  same  sign.  In  discussing  the  conversion  of 
heat  into  work  in  this  article,  we  shall  not  impair  the 
generality  of  our  reasoning  if  we  suppose  that  qa  and  q^ 

are  both  positive ;  in  consequence  of  which  the  ratio  — 

cannot  be  greater  than  -£ . 
Ob 

If  6 a  be  greater  than  #6,  the  body  A  is  said  to  be  at  a 
higher  temperature  than  B.  Hence  we  see  that,  in  a 
complete  reversible  cycle,  a  positive  quantity  of  heat  may 


118  ELEMENTARY  THERMODYNAMICS. 

be  taken  from  any  body  A  and  partially  converted  into 
work  provided  that  there  is  a  second  body  B,  of  lower 
temperature,  to  absorb  the  waste  heat.  The  hotter  body 
A  is  known  as  the  '  source '  and  the  cooler  body  B  as  the 
'  refrigerator.' 

The  heat  which  is  converted  into  work  is  qa  —  qb  and 
its  ratio  to  the  heat  absorbed  from  the  hotter  body  is 

2«_J?&     which,  since  the  cycle  is  reversible,  is  equal  to 

(jo, 
f\ 

1  -  /  .     This  ratio,  which  is  called  the  '  efficiency'  of  the 

"a 

cycle,  depends  therefore  only  on  the  temperatures  of  the 
source  and  refrigerator. 

If  the  cycle  be  not  reversible,  the  ratio    -  will  either 

e         q" 

be  less  than,  or  at  most,  equal  to  -^ .     It  will  be  easily 

seen  that  if  the  irreversible  cycle  be  non-frictional.       will 

6  * 

be  equal  to  -£ ,  and  that  in  all  other  cases  it  will  be  less. 

Vb 

Thus,  when  a  cycle  is  factional,  its  efficiency,  which  is  the 

ratio  ^— ^     or  i_9b     will  be  lesg  than  x  _  fa     the 

9a  qa  6a 

efficiency  of  a  non-frictional  cycle  working  between  the 
same  source  and  the  same  refrigerator.  In  popular 
language,  no  cycle  can  transform  so  large  a  proportion 
of  heat  into  work  as  a  non-frictional1  cycle. 

If  6 a  be  less  than  6b,  qa  will  also  be  less  than  qb.  In 
this  case,  in  the  course  of  a  complete  reversible  cycle,  a 

1  Here,  and  in  much  that  follows,  the  term  '  friction '  is  used  in 
a  general  way  to  include  every  kind  of  irreversibility  as  well  as  friction 
proper. 


CARNOT'S  PRINCIPLE.  119 

positive  quantity  of  beat  is  taken  from  the  cooler  body 
and  a  larger  quantity  given  up  to  the  hotter.  Thus  an 
expenditure  of  work  is  necessary  during  a  reversible  cycle 
if  we  wish  to  lift  a  quantity  of  heat  from  a  lower  to  a 
higher  temperature,  and  the  amount  of  work  required  is 
the  same  for  all  material  systems  and  for  all  reversible 

cycles.     When  the  cycle  is  not  reversible,  —  cannot  be 

y  qb 

/\ 

greater  than  -^ .     Hence  if  qa  have  the  same  value  as  in 

a  reversible  cycle,  qb  will  have  a  greater,  or,  in  the  case  of 
non-frictional  irreversible  cycles,  an  equal  value.  Con- 
sequently, heat  cannot  be  raised  from  a  lower  to  a  higher 
temperature  with  so  small  an  expenditure  of  work  as  when 
the  cycle  is  non-frictional. 

It  is  not  the  least  important  consequence  of  Thomson's 
system  of  temperature  that  heat  cannot  flow  of  itself, 
either  by  conduction  or  radiation,  from  a  lower  to  a 
higher  temperature.  But  when  a  thermometer,  gradu- 
ated in  the  usual  way,  is  employed  to  show  differences 
of  temperature,  the  thermometric  substance  may  be  such 
that  heat  sometimes  appears  to  pass  from  a  colder  to  a 
hotter  body.  For  example,  if  within  certain  limits  of 
temperature  the  thermometric  substance  contracts  as 
its  temperature,  as  shown  by  the  air  thermometer,  in- 
creases, like  water  between  0°  C.  and  4°  C.,  and  if  A,  B 
be  any  two  bodies  whose  temperatures  lie  within  these 
limits ;  then  if  the  temperature  of  A,  as  shown  by  the  air 
thermometer,  be  higher  than  that  of  B,  heat  will  be  able 
to  pass  of  itself  from  A  to  B  and  yet  the  temperature  of 
B  would  be  shown  by  the  thermometer  we  have  just 
described  to  be  higher  than  that  of  A. 


120  ELEMENTARY   THERMODYNAMICS. 

52.  It  has  hitherto  been  usual  to  consider  only  the 
case  in  which  the  material  system  consists  of  a  single 
unelectrified    body   without    mechanical    motions   whose 
temperature  when  in  a  state  of  equilibrium  is  the  same 
throughout.     Such  a  body  when  it  undergoes  a  reversible 
cycle  of  any  kind  is  known  as  a  '  Carnot's  perfectly  re- 
versible engine.' 

In  Carnot's  time,  heat  was  believed  to  be  a  material 
substance  which  could  neither  be  created  nor  destroyed, 
and  the  total  quantity  of  heat  contained  in  any  system  was 
supposed  to  depend  only  on  its  state.  It  thence  followed 
that  in  any  cyclical  process,  exactly  as  much  heat  was 
absorbed  as  given  out,  and  it  was  thought  that  the  work 
obtained  from  the  cycle  was  done  at  the  expense  of  the 
fall  of  temperature ;  just  as  the  work  done  by  a  stream  in 
driving  a  water-wheel  depends  on  a  change  in  the  level  of 
the  stream.  Yet  notwithstanding  these  false  assumptions, 
it  was  easily  shown,  as  in  Article  (49),  that  if  W  be  the 
work  obtained  from  the  engine  during  a  complete  cycle, 
and  Q  the  heat  absorbed  from  the  source  or  given  to  the 

W 

refrigerator,  the  ratio    „   was  the  same  for  all  reversible 

engines  working  between  any  sources  and  refrigerators 
whose  temperatures  were  respectively  equal  to  those  of  two 
given  bodies  A,  B.  It  could  therefore  depend  only  on  the 
temperatures  of  A  and  B.  In  addition  to  this  conclusion, 
Carnot  also  obtained  one  of  the  most  important  results  of 
the  next  chapter. 

53.  Carnot's  principle  includes  two  of  the  three  ex- 
perimental results  given  in  elementary  books  as  The  laws 
of  Friction.     The  third  cannot  be  deduced  from  Carnot's 


CARNOT'S  PRINCIPLE.  12l 

principle,   because   it   is   not   rigidly   accurate,   although 
generally  a  very  good  approximation. 

Suppose  that  we  have  two  bodies  A,  B,  of  the  same 
temperature,  which  possess  no  potential  energy,  or  at 
least,  only  a  constant  amount,  and  let  them  be  rubbed 
together  by  the  hand  in  such  a  way  that  the  other  ex- 
ternal forces  do  no  work.  Then  if  the  mechanical  motions 
be  zero  both  at  the  beginning  and  at  the  end  of  the 
operation,  the  cycle  may  be  completed  by  allowing  the 
two  bodies  to  exchange  heat  with  some  third  body  C. 
Hence,  by  Carnot's  axiom,  there  is  an  expenditure  of 
mechanical  work  during  the  cycle,  and  therefore  we  con- 
clude that : — 

I.  Friction  acts  on  each  body  in  a  direction  opposite 
to  that  in  which  its  relative  motion  takes  place,  or  merely 
tends  to  take  place. 

II.  No  more  friction  can  ever  be  called  into  play 
than  is  just  sufficient  to  prevent  relative  motion. 

54.  It  is  sometimes,  but  erroneously,  supposed  that 
Carnot's  principle  asserts  that  it  is  impossible  to  raise  the 
temperature  of  a  body  without  an  expenditure  of  work. 
The  following  two  examples  will  illustrate  this  and  similar 
points. 

(1)  If  the  bulb  of  a  thermometer  be  wrapped  round 
with  flannel  and  then  breathed  on,  the  temperature  of  the 
thermometer  is  found  to  rise  higher  than  the  temperature 
of  the  breath.  This  experimental  fact  will  occasion  no 
difficulty  if  we  remember  that  Carnot's  axiom  asserts 
nothing  unless  the  changes  of  state  constitute  a  com- 
plete cycle,  and  that  even  then,  the  only  temperatures 
taken  into  account  are  the  temperatures  of  the  two 


122 


ELEMENTARY   THERMODYNAMICS. 


bodies  from  which  heat  is  received  or  to  which  heat 
is  lost. 

(2)  The  following  ingenious  operation  was  described 
by  Hirn  in  1862.  Let  there  be  two  similar  and  equal 
cylinders  surrounded  by  good  non-conducting  materials 
and  connected  at  the  bottom  by  a  very  narrow  copper 
pipe.  Also  let  the  cylinders  be  fitted  with  air-tight 
pistons  the  rods  of  which  are  constructed  with  equal  teeth 
to  engage  a  spur  wheel  so  that  when  one  piston  rises  the 
other  will  descend  an  equal  distance.  The  whole  space 
beneath  the  pistons  will  then  be  invariable,  because,  as 
the  space  under  one  piston  decreases,  the  space  under  the 
other  increases  by  an  equal  amount. 

Now  let  us  suppose  that  initially  the  piston  B  is  at 
the  bottom  of  its  cylinder  and  the  other  cylinder  filled 


with  gas  at  0°  C.,  the  pipe  being  kept  surrounded  by  the 


CARNOT'S  PRINCIPLE.  123 

steam  of  boiling  water.  Then  let  the  wheel  be  slowly 
turned  so  that  the  piston  A  is  caused  to  descend  and  B  to 
rise.  If  the  pistons  be  well  made,  very  little  mechanical 
work  will  be  required  but  important  effects  will  be  pro- 
duced on  the  gas.  For  as  soon  as  the  motion  commences, 
a  small  quantity  of  gas  will  pass  through  the  pipe  where 
its  temperature  will  be  raised  from  0°  C.  to  100°  C.  In 
consequence  its  volume  will  increase  and  the  gas  which 
remains  in  A  will  be  somewhat  compressed  and  its  tempe- 
rature slightly  raised.  As  the  next  small  quantity  of  gas 
passes  through  the  pipe,  it  will  be  heated  and  will  expand 
in  the  same  way,  and  thereby  the  gas  contained  in  both 
cylinders  will  be  still  further  compressed  and  raised  in 
temperature.  The  motion  of  the  wheel  being  continued 
until  the  piston  A  gets  to  the  bottom  of  the  cylinder,  the 
whole  of  the  gas  will  have  passed  through  the  pipe  into 
the  other  cylinder  and  the  temperature  of  every  part  will 
be  100°  C.  or  higher,  the  mean  temperature  being  about 
120°  C.  Hence  in  completing  the  cycle  by  bringing  the 
temperature  of  every  part  of  the  gas  back  to  0°  C.,  heat 
may  pass  out  of  the  gas  to  bodies  whose  temperatures  are 
considerably  above  100°  C.  We  are  therefore  able  to 
raise  heat  from  a  lower  to  a  higher  temperature  without 
an  expenditure  of  work.  But  it  must  be  remembered 
that  Carnot's  principle  supposes  that  there  are  only  two 
external  bodies  with  which  the  system  can  exchange  heat. 
In  the  present  instance,  the  cycle  cannot  be  completed 
unless  there  are  bodies  of  as  low  a  temperature  as  0°  C.  to 
which  the  gas  may  part  with  its  heat.  Examples  like 
this  cannot  be  fully  discussed  until  we  have  considered 
the  case  in  which  there  are  any  number  of  external  bodies 
with  which  a  given  system  can  exchange  heat. 


124  ELEMENTARY  THERMODYNAMICS. 

55.  When  any  material  system  is  subjected  to  ex- 
ternal electric  influences,  it  is  easily  proved  in  works  on 
electricity  that   the   system   may  gain   or   lose   'electric 
energy'  (electric  work)  without  either  gaining  or  losing 
heat.     If  the  system  undergo  a  cyclical  process  during 
which  there  is  neither  absorption  nor  evolution  of  heat, 
the  whole  of  the  work  obtained  during  the  cycle  will  have 
been  transformed  out  of  electric  energy,  so  that  electric 
energy  may  be  transformed  wholly  into  mechanical  work, 
and,  conversely,   mechanical   work   may   be   transformed 
wholly  into   electric    energy.      Hence,  by  Carnot's  prin- 
ciple, electric   energy  may  be    transformed   wholly   into 
heat,  but    heat   can   only  be   transformed  partially  into 
electric  energy. 

Any  system  which  absorbs  or  loses  electric  energy 
may  be  called  an  electric,  or  an  electro-magnetic,  engine, 
and  it  is  evident  that  it  is  not  subject  to  the  restrictions 
of  Carnot's  principle.  For  the  future,  we  shall  suppose 
that  there  are  no  external  electric  influences,  so  that 
Carnot's  principle  will  always  be  true. 

56.  When  any  material  system  is  enclosed  in  a  vessel 
of  any  kind  which   is   protected    from   all    external   in- 
fluences, mechanical,   thermal,   and   electrical   (including 
the  radiation  of  energy  into  external  space  as  an  external 
influence),  it  may  for  the  present  be  assumed  as  an  ex- 
tension of  Carnot's  principle,  or  of  the  principle  of  friction, 
that  the  system  will  ultimately  attain  a  state  in  which  all 
its  visible  properties  are  invariable  and  the  temperatures 
of  all  its  parts  equal.     The  whole  system  will  then  behave 
as  a  rigid  body,  so  that  the  only1  mechanical  motions  it 

1  See  Art.  24. 


CARNOT'S  PRINCIPLE.  125 

can  have  consist  of  a  uniform  motion  in  a  straight  line  of 
the  centre  of  mass  combined  with  a  constant  angular 
rotation  of  the  whole  system  about  some  straight  line 
passing  through  the  centre  of  mass  in  a  constant  direc- 
tion. Also  since  radiation  depends  011  the  non-mechanical 
motions  which  always  exist  except  when  the  absolute 
temperature  is  zero,  it  may  be  further  assumed  that  if 
the  given  system  had  been  at  liberty  to  radiate  energy 
into  infinite  space  but  had  otherwise  been  protected  as 
before,  it  would  still  have  ultimately  attained  an  invari- 
able state,  bat  the  final  absolute  temperature  of  every 
part  would  have  been  zero. 

It  should  be  observed  that  if  the  system  consist  of 
several  parts  the  temperatures  of  which  are  originally 
different,  it  is  not  asserted  that  the  colder  parts  cannot 
become  as  hot  as  the  hotter  until  the  final  state  is 
reached.  Suppose,  for  example,  that  the  system  consists 
of  an  unelectrified  part  A  at  130°  C.  and  an  unelectrified 
part  B  at  70°  C.,  both  parts  being  without  mechanical 
motions,  and  let  .B  be  a  cylinder  in  which  a  quantity  of 
gas  is  confined  by  an  air-tight  piston  pressed  upon  by  a 
strong  spring  but  prevented  from  being  forced  in  by  a 
fusible  plug  which  melts  at  94°  C.  Then  if  A  and  B 
be  placed  in  presence  of  one  another  in  a  vessel  which 
is  protected  from  all  external  influences  and  has  so 
small  a  capacity  for  heat  that  its  cooling  or  heating 
effects  on  A  and  B  are  very  small,  it  is  clear  that  A 
will  cool  and  B  get  hotter  until  the  melting  of  the 
fusible  plug  releases  the  spring.  When  this  occurs, 
the  temperature  of  B  may  rise  to  200°  C.  or  more, 
and  then  B  would  begin  to  cool  and  A  would  get 
hotter. 


126  ELEMENTARY   THERMODYNAMICS. 

57.  When  a  quantity  of  radiation  falls  on  a  body,  it 
is  easily  shown  by  experiment  that  part  of  the  radiation 
may  be  absorbed,  part  reflected,  and  the  remainder  trans- 
mitted ;  but  since  the  tendency  to  equalisation  of  tempe- 
rature is  universal,  it  is  evident  that  no  body  can  reflect 
or  transmit  the  whole  of  the  radiation  which  falls  upon 
them.  There  is  nothing  in  Carnot's  principle,  however,  to 
prevent  us  from  thinking  that  a  body  may  absorb  the 
whole  of  the  radiation  which  falls  upon  it,  and  in  fact, 
any  body  well  coated  with  lamp-black  practically  does  so. 
We  shall  accordingly  suppose  that  a  surface  of  lamp-black 
absorbs  the  whole  of  the  incident  radiant  energy.  This 
assumption  is  made  merely  as  a  convenience  and  not  to 
prove  any  properties  which  could  not  be  proved  without 
it. 

Again,  if  a  piece  of  ice  be  brought  near  the  bulb  of  a 
thermometer,  the  temperature  of  the  thermometer  is 
observed  to  fall;  but  when  a  piece  of  red-hot  iron  is 
brought  near,  the  temperature  rises.  These  facts  have 
led  to  Prevost's  theory  which  has  been  fully  developed  by 
Stewart  and  others  under  the  name  of  the  Theory  of 
Exchanges.  It  is  supposed  that  every  system  is  constantly 
radiating  energy  at  a  rate  which  depends  only  on  its  state 
and  is  independent  of  the  presence  or  absence  of  other 
systems.  In  the  two  cases  just  given,  heat  is  radiated 
from  the  thermometer  faster  than  it  is  received  from  the 
ice  and  not  so  fast  as  it  is  received  from  the  red-hot  iron. 
In  applying  Carnot's  principle  to  radiation,  we  shall 
restrict  ourselves  to  the  case  of  unelectrified  bodies 
without  mechanical  motions.  The  body  B  which  is  to 
be  considered  will  be  supposed  suspended  by  a  fine 
thread  or  string  in  a  closed  vessel  V,  of  any  shape  or 


CARNOT'S  PRINCIPLE.  127 

size,  from  which  the  air  is  exhausted,  so  that  radiation 
can  only  pass  between  the  body  B  and  the  containing 
vessel  V  across  a  vacuum ;  and  it  will  be  assumed,  as 
usual,  that  radiant  energy  travels  in  a  vacuum  in  straight 
lines  in  directions  which  can  neither  be  deflected  nor 
reversed  except  by  some  material  obstacle.  We  shall  also 
suppose  that  the  vessel  V  is  protected  from  all  external 


influences,  so  that  V  and  B  ultimately  assume  the  same 
constant  uniform  temperature. 

When  B  has  attained  its  final  state,  it  is  evident  that 
it  must  be  absorbing  exactly  as  much  radiation  as  it 
emits.  But  if  Q  be  the  quantity  of  radiation  which  falls 
on  B  per  second,  one  part,  qr,  will  be  reflected,  another 
part,  qt,  transmitted,  and  the  remainder,  qa,  absorbed, 
where 

qr  +  qt  +  qa  =  Q, 

so  that  the  amount  of  absorption  or  emission  per  second 
is  given  by 

qa  =  Q-qr-qt, 
all  the  quantities  being  positive. 

Now  if  B  had  been  coated  with  lamp-black,  qr  and  qt 
would  both  have  been  zero,  and  the  emission  per  second 
would  have  been  equal  to  Q,  which  is  greater  than  in  any 
other  case.  This  result  is  usually  expressed  by  saying 


128  ELEMENTARY   THERMODYNAMICS. 

that  good  reflectors  or  transmitters  are  bad  radiators.  It 
must  be  remembered,  however,  that  this  property  is  only 
shown  to  be  true  when  B  has  attained  its  final  state : 
thus  if  B  reflect  well  at  50°  C.,  it  does  not  necessarily 
radiate  badly  at  all  other  temperatures. 

Again,  since  the  radiation  emitted  by  B  per  second 
when  its  surface  is  coated  with  lamp-black  is  equal  to  the 
whole  radiation  which  falls  upon  it,  it  is  clear  that  the 
incident  radiation  in  this  case  is  independent  of  the 
position  of  B  inside  F,  and  also  of  the  nature  and  shape 
and  size  of  V. 

58.  When  a  body  is  coated  with  lamp-black,  the 
emission  of  radiation  will  be  a  surface  phenomenon,  be- 
cause all  the  radiation  coming  from  the  interior  will  be 
absorbed  by  the  lamp-black  surface.  Hence  the  emission 
per  unit  of  area  per  second  at  any  point  of  the  surface 
will  be  independent  of  the  curvature  of  the  surface  at 
that  point.  Consequently,  if  dq  be  the  radiant  energy 
emitted  into  a  vacuum  per  unit  of  area  per  second  at  any 
point  P  of  the  surface  within  a  small  right  cone  of  solid 
angle  day  whose  axis  makes  an  angle  6  with  the  normal  at 

P,  the  limiting  value  of  -~  will  be  a  function  of  6  only. 

If  therefore  da-  be  a  small  area  of  the  surface  at  P,  the 
radiation  sent  out  into  a  vacuum  by  this  area  within  the 
small  cone  in  a  second  will  be 

d<Tda>f(e}  (30). 

From  this  result  we  may  obtain  some  very  important 
conclusions. 

(1)  Let  us  take  a  closed  vessel  V  whose  interior  is  in 
the  form  of  a  hemisphere  of  radius  I  with  a  fine  circular 


CARNOT'S  PRINCIPLE.  120 

cylindrical  recess,  the  axis  of  which  is  at  right  angles  to 
the  base  of  the  hemisphere  and  passes  through  the  centre 
0,  as  in  the  figure.  Let  the  recess  be  nearly  filled  by  a 
plug  B,  one  of  whose  ends  is  in  the  same  plane  as  the 
base  of  the  hemisphere.  Also  let  both  the  interior  of  V 
and  the  whole  of  B  be  coated  with  lamp-black  and  then 
exhaust  the  air  from  V. 


Then  if  any  line  OP  be  drawn  from  0  to  the  surface 
of  the  hemisphere  making  an  angle  <£  with  the  perpen- 
dicular Oz  through  0  to  the  base,  and  if  da.  be  the  area 
of  the  end  of  B,  the  solid  angle  it  subtends  at  P  will  be 

da.  cos  <f> 

~Ji       ' 

Hence  the  radiation  sent  per  second  in  the  direction  of 
the  end  of  B  by  that  part  of  the  hemispherical  surface 
included  between  the  two  right  cones  (f),<f>  +  d(f>,  described 
about  Oz,  will  be 


or  2-7T/  (0)  da  sin  <£  cos  (f>d(f>. 

Of  this  radiation,  the  whole  may  reach  B  or  part  may  be 
lost  by  friction  in  the  ether.  It  is  therefore  necessary  to 
denote  the  part  which  falls  on  B  by 

F  [I,  27r/(0)  da,  sin  $  cos  <f>d<f>], 
P.  9 


130 


ELEMENTARY    THERMODYNAMICS. 


or,  expanding  in  powers  of  da  by  Maclaurin's  theorem, 

27r/(0)  doi-^r  (I)  sin  <£  cos  <j>dtf>, 

where  it  is  obvious  that  ^  (I)  can  have  no  constant  value 
but  unity. 

Thus  the  total  radiation  which  falls  on  the  end  of  B  per 
second  is 


f 


sin  <  cos 


or  7r/(0)  dcL-^r  (I). 

This  is  equal  to  the  emission  from  the  end  of  B  per 
second,  and  is  therefore  independent  of  I  :  thus  ty  (I)  =  a 
constant  and  the  ether  is  frictionless. 

(2)  Again,  suppose  that  F  and  B  are  concentric 
blackened  spheres  of  radii  R  and  r,  respectively,  and 
let  the  air  be  exhausted  from  F,  as  before.  Then  the 
radiation  sent  to  B  per  second  by  each  unit  of  area  of  F 


where 


CARNOT'S  PRINCIPLE.  131 

But  the  radiation  emitted  by  B  is  proportional  to  its  area, 
and  equal  to  InrXr2,  say.     Hence 


or  27r  (  V(0)  sin  BAG  =  X  sin2  0,. 

J  o 

Differentiating, 

27r/(01)  sin  0j  =  2X  sin  ^  cos  0,  , 
so  that  /(0)  =  /iCos0, 

where  /t  is  a  constant. 

Thus  the  emission  into  a  vacuum  per  second  by  a  small 
area  da-  within  the  cone  dw  takes  the  form 

H  cos  Odadw  ..................  (31). 

Consequently  the  radiation  sent  out  into  a  vacuum  per 
unit  of  area  per  second  at  any  point  P  of  the  blackened 
surface  within  a  right  cone  described  about  the  normal 

at  P  with  any  finite  semi-vertical  angle  0,  f  <  •=  J  is 

27r/i  f  6l  sin  6  cos  6de  =  TTH  sin2  6l  ......  (32), 

J  o 

and  the  total  radiation  at  P  is  TT/A.     Calling  this  k,  the 
emission  within  the  cone  da  becomes 
k 

-  COS  0d(i). 
7T 

(3)  Since  radiant  energy  travels  in  a  vacuum  in 
straight  lines,  if  two  bodies  A,  B,  be  separated  by  a 
vacuum,  it  will  only  be  possible  for  one  ray  to  pass 
from  a  point  P  of  one  to  a  point  Q  of  the  other.  But 
if  a  material  body  L  be  interposed  between  A  and  B,  it 
may  happen  that  several  of  the  rays  which  proceed  from 

9—2 


132 


ELEMENTARY   THERMODYNAMICS. 


P  have  their  directions  so  bent  in  passing  through  L  that 
they  are  concentrated  on  a  very  small  area  surrounding 


the  point  Q.  One  of  the  best  examples  of  this  is  seen 
when  the  rays  of  the  sun  are  brought  to  a  focus  by  a  lens 
(of  glass  or  ice).  If  the  lens  is  large  enough,  the  effect 
produced  may  be  considerable. 

It  will  now  be  naturally  asked  whether  it  would  not 
be  possible,  by  means  of  a  system  of  lenses,  to  cause  heat 
to  pass  from  a  cooler  to  a  hotter  body  without  an  ex- 
penditure of  mechanical  work,  as  was  thought  by  Rankine? 
A  careful  examination  of  the  question  by  Kirchhoff  and 
Clausius  has  led  them  to  the  conclusion  that  such  a  result 
is  impossible.  To  give  a  simple  illustration,  let  us  suppose 
two  bodies  A,  B,  and  a  lens  L  to  be  situated  in  a  vacuum; 
and  let  us  neglect  the  reflection  and  absorption  that  take 
place  as  the  rays  pass  through  L.  Then  if  the  focal 


length  of  the  lens  be  large,  the  rays  which  fall  on  it  from 
.1  distant  point  P  will  converge  with  considerable  accuracy 


CAENOT'S  PRINCIPLE.  133 

to  a  point  Q ;  and  if  PQ  be  the  principal  axis  of  the  lens, 
the  rays  from  any  point  p  of  a  small  area  dS,  placed  at  P 
at  right  angles  to  PQ,  will,  after  passing  through  the  lens, 
converge  to  a  point  q  of  a  small  area  ds,  placed  at  Q 
parallel  to  dS,  the  lines  pq,  PQ  both  passing  through  C, 
the  optical  centre  of  the  lens. 


Now  if  R  be  the  distance  CP  and  r  the  distance  CQ,  we 
shall  have 


Also  the  rays  sent  to  the  lens  by  any  point  p  of  dS  and 
by  any  point  q  of  ds,  form  small  right  cones  about  the 
normals  at  p  and  q  whose  semi-  vertical  angles  are  as  r  to 
R.  Hence  if  for  simplicity  we  suppose  dS  and  ds  to  be 
•coated  with  lamp-black  (so  that  there  is  no  reflection),  we 
see  that  the  rate  at  which  radiation  is  sent  by  dS  to  ds 
through  the  lens  is  equal  to  the  rate  at  which  it  is  sent 
by  ds  to  dS. 

59.  Advantage  has  been  taken,  by  Sir  J.  Herschell 
and  Pouillet,  of  the  peculiar  property  possessed  by  lamp- 
black of  absorbing  practically  all  the  radiation  which  falls 
upon  it,  to  estimate  the  amount  of  energy  radiated  by  the 
sun  to  the  earth.  Of  course,  it  is  only  possible  to  measure 
the  quantity  that  actually  reaches  the  surface  of  the  earth  ; 
but  the  amount  prevented  from  reaching  the  surface  by 


134  ELEMENTARY   THERMODYNAMICS. 

the   atmosphere  can  be  roughly  allowed  for  by  taking- 
observations  when  the  sun  has  different   altitudes,  and 


Sun. 


Air. 


Earth. 

when,  in  consequence,  different  thicknesses  of  air  have  to 
be  traversed,  as  shown  in  the  figure.  It  thus  appears 
probable  that  only  one  half  of  the  radiant  energy  sent 
by  the  sun  to  the  earth  succeeds  in  reaching  the  surface. 
Taking  this  into  account,  it  is  calculated  that  the  quantity 
of  radiation  which  falls  on  the  earth  in  a  year  would  be 
sufficient  to  melt  a  layer  of  ice  30  metres  (100  feet)  thick, 
covering  the  entire  surface. 

Since  a  gramme  of  ice  at  0°  C.  occupies  1  '087  cubic 
centimetres,  and  since  the  latent  heat  of  ice  is  79'25 
calories,  the  total  radiation  which  falls  on  the  earth  from 
the  sun  in  a  year  is  equivalent  to  about  3000  x  73  calories, 
or  9  x  1012  ergs,  per  square  centimetre.  This  is  about  the 
same  as  25  calories,  or  109  ergs,  per  square  centimetre  per 
hour ;  or  again,  3  x  105  ergs  (^  kilogrammetre)  per 
square  centimetre  per  second.  In  English  measure,  the 
average  quantity  of  radiation  per  square  foot  which 
actually  reaches  the  surface  of  the  earth  in  an  hour,  is 
sufficient  to  raise  the  temperature  of  25£  Ibs.  of  water 
from  0°C.  to  1°  C.,  which  is  equivalent  to  35,500  foot- 
pounds. 

Now  if  a  particle  receives  a  small  displacement  of  ds 
centimetres  whilst  it  is  acted  on  by  a  force  whose  resolved 


CARNOT'S  PRINCIPLE.  135 

part  in  the  direction  of  the  displacement  is  p  dynes,  the 
work  done  by  the  force  on  the  particle  is  pds  ergs.  Con- 
sequently, if  the  time  of  describing  the  elementary  path 
ds  be  dt  seconds,  the  rate  at  which  work  is  done  by  the 

force  on  the  particle  is  p  -v, ,  or  pv,  ergs  per  second.     If 

therefore  we  divide  the  rate  at  which  work  is  being  done 
by  the  velocity  of  the  particle,  we  shall  get  the  resolved 
part  in  the  direction  of  motion  of  the  force  which  acts 
upon  it.  Suppose  then,  merely  for  the  purpose  of  the 
following  illustrative  numerical  calculation,  that  each 
particle  of  a  body  possesses  the  same  non-mechanical 
velocity  and  let  the  mechanical  motions  be  so  small  that 
they  need  not  be  taken  into  account.  Then  if  we  divide 
the  rate  at  which  radiation  is  being  gained  by  the  common 
non-mechanical  velocity  of  each  particle,  we  shall  obtain 
the  sum  of  the  resolved  parts  in  the  direction  of  motion 
of  the  radiation  forces  which  act  on  the  several  particles, 
or,  as  we  may  call  it,  the  total  radiation  force  acting  on 
the  body.  Thus  if  we  take  the  non-mechanical  velocity 
of  each  particle  of  the  earth's  crust  to  be  50,000  centi- 
metres per  second,  and  suppose  the  total  gain  of  radiation 
by  the  crust  to  be  x  times  the  radiation  which  actually 
reaches  it,  the  total  radiation  force  per  square  centimetre 
will  be  3#  dynes ;  which  is  about  the  same  as  SQ'Gx 
grammes  per  square  metre.  In  English  measure,  the  total 

radiation  force  per  square  foot  would  be  about  -j-w^  Ib. 

Of  course,  the  non -mechanical  motions  really  take 
place  in  such  different  directions  that,  in  the  case  of  any 
finite  area,  the  resultant  of  all  the  radiation  forces,  found 
by  the  ordinary  rules  of  statics,  will  generally  be  quite 


136  ELEMENTARY   THERMODYNAMICS. 

insignificant.  When  a  material  system  has  attained  its 
final  state,  as  explained  in  Art.  56,  the  resultant  will  be 
strictly  zero  ;  but  when  the  temperatures  of  different  parts 
of  a  body  are  unequal  and  undergoing  rapid  variations, 
this  is  not  necessarily  the  case.  Thus,  for  example,  the 
orbit  of  a  comet  is  generally  in  the  form  of  a  very 
elongated  ellipse  with  the  sun  in  one  focus,  and  when 
the  comet  is  describing  that  part  of  the  orbit  nearest 
the  sun,  its  velocity  becomes  very  great.  Hence,  since 
the  intensity  of  solar  radiation  evidently  varies  inversely 
as  the  square  of  the  distance  from  the  sun's  centre,  as 
the  comet  rushes  close  past  the  sun,  the  intensity  of  the 
radiation  which  falls  upon  it  will  increase  with  enormous 
rapidity  to  a  maximum  and  then  decrease  as  fast.  Under 
these  conditions,  it  would  scarcely  be  possible  for  the 
radiation  forces  which  act  on  the  comet  to  be  so  delicately 
balanced  as  to  produce  no  visible  mechanical  effect.  Per- 
haps the  tails  of  comets  may  be  due  to  this  cause  ;  for 
they  appear  to  be  produced  by  a  repulsive  force  residing 
in  the  sun,  and  we  know  that  the  force  required  is  generally 
small,  the  mass  of  a  comet's  tail  being  sometimes  under 
100  grammes  (3'5  ounces). 

If  a  be  the  radius  of  the  earth  and  d  its  distance  from 

the  sun,  the  total  energy  radiated  by  the  sun  is 

TTtt2 

times  as  much  as  falls  on  the  earth  in  the  same  time. 


Since      =  23000,  we  find    ~  =  2100  million.     Hence  if 

6  be  the  radius  of  the  sun,  the  radiation  emitted  by  the 
sun  per  hour  is  sufficient  to  melt  a  layer  of  ice  covering 

its  entire  surface  21  x  108  x  x  6—  ?^  --  metres  thick. 

3  60  x  24 


CARNOT'S  PRINCIPLE.  137 

Substituting  110  for  -  ,  we  find  this  thickness  to  be  about 
600   metres  (660  yards).     Again,  the  radiation  from   a 

given  area  of  the  sun  is  21  x  108  x     '      =  175000,  times 

- 


as  much  as  falls  on  an  equal  area  of  the  earth  in  the  same 
time.  It  is  therefore  equal  to  4,375,000  calories  per  square 
centimetre  per  hour;  or  to  1215  calories,  or  525  x  108 
ergs,  per  square  centimetre  per  second.  In  English 
measure,  the  radiation  per  hour  for  each  square  foot  is 
sufficient  to  raise  the  temperature  of  90,000  Ibs.  of  water 
from  0°C.  to  100°  C. 

If  distances  be  expressed  in  centimetres,  we  have 
a  =  64  x  107,  and  therefore  the  total  radiation  emitted 
by  the  sun  in  a  year  is 

(21  x  108)  x  (4<7ra2  x  9  x  1012)  =  1041  ergs. 

60.  It  may  be  taken  that  the  furnace  of  a  locomotive 
consumes  1  Ib.  of  coal  per  minute  for  each  square  foot  of 
grate,  and  that  each  pound  of  coal  burnt  gives  out  suf- 
ficient heat  to  raise  the  temperature  of  75  Ibs.  of  water 
from  0°C.  to  100°C.  The  radiation  from  a  given  area 
of  the  sun  is  therefore  20  times  the  heat  generated  in 
the  same  time  in  a  locomotive  furnace  with  the  same 
area  of  grate.  In  making  the  comparison,  it  must,  how- 
ever, be  borne  in  mind  that  while  the  furnace  of  a  loco- 
motive is  of  very  small  depth,  the  radiation  emitted  by 
the  sun  is  not  a  mere  surface  phenomenon,  but  probably 
comes  to  a  large  extent  from  considerable  distances  below. 
In  fact,  while  the  absolute  temperature  of  the  fire  of  a 
locomotive  may  be  as  high  as  1800°,  the  temperature  of 
the  sun's  surface  is  only  estimated  at  from  3000°  to 


138  ELEMENTARY   THERMODYNAMICS. 

10000°.     The  central  parts  of  the  sun  will,  of  course,  be 
at  a  higher  temperature. 

61.  Proceeding  now  to  a  very  important  extension 
of  Carnot's  axiom,  let  us  take  any  material  system  which 
may  possess  any  mechanical  motions  and  may  be  electrified 
in  any  way  we  please,  but  is  not  subjected  to  external 
electric  influences;  and  suppose  it  made  to  undergo  a 
cyclical  process  during  which  it  is  at  liberty  to  lose  or 
gain  heat  in  any  possible  way.  Whilst  the  temperature  6 
at  any  point  P  of  the  system  varies  continuously  by  an 
infinitesimal  amount  dd  in  the  time  dt,  let  the  small 
quantity  of  heat  absorbed  near  the  point  P,  either  by 
conduction  or  radiation,  be  denoted  by  dtdh.  Whether 
the  point  P  be  on  the  surface  or  far  in  the  interior,  the 
heat  dtdh  may  be  supposed,  by  a  stretch  of  the  imagina- 
tion, to  be  supplied  by  contact  with  a  vessel  filled  with  a 
liquid  and  its  vapour.  If  the  liquid  be  sufficiently  volatile, 
the  temperature  will  be  uniform  throughout  the  vessel ; 
for  a  difference  of  temperature  in  any  part  would  cause 
an  explosive  formation  or  condensation  of  vapour  until 
the  temperature  became  uniform.  Also  if  the  sides  of 
the  vessel  be  supposed  to  be  very  good  conductors  and 
to  have  an  exceedingly  small  capacity  for  heat,  the  tem- 
perature of  the  liquid,  when  the  vessel  is  in  contact  with 
any  part  of  the  system,  will  be  the  same  as  the  tempera- 
ture of  that  part  of  the  system  with  which  the  vessel  is 
in  contact.  We  are  therefore  at  liberty  to  suppose  that 
all  the  heat  dtdh  is  imparted  by  a  'Carnot's  perfectly 
reversible  engine'  which  continually  brings  it  from  a 
source  whose  temperature  is  uniform  and  constantly 
equal  to  00.  The  heat  dq0  taken  from  the  source  to 


CARNOT'S  PRINCIPLE.  139 

supply  a  quantity  of  heat  dtdh  at  the  temperature  0 
being  given  by  the  relation 

dq0  _  dt  dh 

e0~  0  ' 

we  have  dq0  =  00dt  -„  . 

The  total  amount  of  heat  taken  from  the  source  in  the 
time  dt  will  therefore  be 

'dh 
0  ' 

where  the  integral  extends  throughout  the  system  at  the 
time  t.  In  a  complete  cycle,  the  total  quantity  of  heat 
taken  from  the  source  will  be 


Hence,  since  in  any  complete  cycle  heat  cannot  be  taken 
from  a  body  whose  temperature  is  uniform  and  constant 
unless  some  other  body  of  different  temperature  be  also 
present,  and  since  00  is  always  positive,  we  infer  that,  for 

a  non-frictional  cycle,  the  integral   Idt  I  -~    is  zero,  and 

that  for  all  other  cycles  it  is  negative. 

If  the  temperature  of  the  system  be  uniform  both  at 
the  beginning  and  at  the  end  of  the  cycle,  these  results 
may  be  more  simply  expressed  by  saying  that  the  integral 

I  —5-  is  zero  for  non-frictional  cycles  and  negative  in  all 

other  cases,  dQ  being  the  heat  absorbed  by  the  whole 
system  whilst  its  temperature  varies  continuously  from 
0  to  6  +  dd,  which  may  take  place  at  different  times  in 
different  parts  of  the  system. 


140  -ELEMENTARY   THERMODYNAMICS. 

62.  The  results  of  the  preceding  article  lead  to  some 
important  conclusions  relating  to  those  states  of  a  system 
through  which  it  can  be  made  to  pass  by  non-frictional 
methods ;  for  if  P,  Q  be  any  two  such  states,  it  may  be 
assumed,  as  a  result  of  observation  and  experience,  that 
the  system  can  be  made  to  pass  from  P  to  Q  by  a  non- 
frictional  process.  Thus  if  a  system  pass  from  an  initial 
state  0  to  a  second  state  A  by  any  non-frictional  path 
OP  A,  and  then  return  from  A  to  0  by  a  non-frictional 
path  AXO,  and  the  corresponding  integrals  be  distin- 
guished by  suffixes,  as  below,  we  have 

fdk 


Since  the  system  in  the  states  0  and  A  is  either  of  uni- 
form temperature  throughout  or  consists  of  a  number  of 
different  bodies  of  different  uniform  temperatures,  the  last 
equation  may  also  be  written 


ff 

J     Q  or  A 


=o, 


where  0  now  denotes  the  uniform  temperature  of  any  one 
of  the  different  bodies  at  any  instant  during  the  process. 
If  the  system  pass  from  0  to  A  by  another  non-frictional 
path  OQA,  and  then  return  by  the  same  path  AXO,  as 
before,  we  have 


Hence 


CARNOT'S  PRINCIPLE.  141 

which  is  the  same  as 


These  important  equations  were  given  by  Sir  \V. 
Thomson  and  Clausius  about  the  same  time.  In  the 
hands  of  Clausius  they  have  led  to  some  further  results 
which  may  be  classed  as  the  most  remarkable  in  the 
whole  range  of  thermo-dynamics.  For  since  the  integrals 
just  obtained  depend  only  011  the  two  states  0,  A,  if  the 
state  0  be  chosen  as  a  standard  fixed  state,  each  integral 
will  depend  only  on  the  state  A,  and  may  be  written  <f>A. 
The  function  </>  was  called  by  Clausius  the  Entropy  of  the 
system  ;  and  since  its  value  would  only  be  altered  by  a 
constant  if  a  different  state  were  taken  as  the  standard 
state,  it  is  clear  that  the  entropy  is  a  single-valued 
function  of  the  independent  variables  which  define  the 
state  of  the  system  together  with  an  arbitrary  additive 
constant  depending  only  on  the  choice  of  the  standard 
state. 

If  the  system  pass  in  a  non-frictional  manner  from  0 
through  A  to  another  state  B,  we  obtain 


or 

which  may  also  be  written 

-*.-*,  ..................  (34)- 


142  ELEMENTARY   THERMODYNAMICS. 

Hence  if  the  system  be  prevented  from  receiving  or  losing 
heat,  all  the  different  states  into  which  it  can  be  brought 
by  non-frictional  processes  have  the  same  entropy.  Such 
non-frictional  processes  are  called  Adiabatic  or  Isentropic 
processes. 

Again,  if  the  temperature  of  every  part  of  the  system 
ke°ep  the  same  constant  value  0,  (34)'  takes  the  simpler 
form 

2/^  =  0  (</>,-</>,)  ...............  (34)". 

When  the  path  AB  is  frictional,  let  the  system  return 
from  B  to  A  by  a  non-frictional  path,  so  as  to  complete 
the  cycle.  Then  by  the  last  article  and  by  equation  (34), 
we  have 


that  is,  dt  <**"^  ...............  (35)> 


Since  the  system  in  the  states  A,  B  consists  of  a  number 
of  bodies  of  uniform  temperatures,  we  obtain 


e 


where  dQ  is  the  heat  absorbed  by  any  part  of  the  system 
while  its  temperature  varies  continuously  from  6  to 

B+M. 

As  no  process  in  nature  can  exactly  be  a  non-frictional 
process,  it  follows  from  equation  (35)  or  (35)',  that  when  a 
system  is  prevented  from  losing  or  gaining  heat,  its 
entropy  must  be  continually  increasing. 

If  the   definition  of  entropy  apply  to  two  states  in- 


CARNOT'S  PRINCIPLE. 


143 


definitely  near  together,  then  when    the   system   passes 
from  one  to  the  other,  we  have 


[dh  = 

dt\-7r< 


.(36), 


.(36)'. 


63.     The  simple  relation  2,   ^  =  d(j>,  which  holds  for 

any  small  non-frictional  change  of  state,  becomes  dQ  =  6d(f> 
when  we  consider  only  a  single  body  of  uniform  tempera- 
ture. Hence  if  we  take  two  rectangular  axes  and  denote 
the  temperature  and  entropy  of  the  body  at  any  instant  by 


MN 


0 


the  ordinate  PM  and  abscissa  OM  of  a  point  P,  the  heat 
absorbed  by  the  body  when  a  small  change  of  state  PQ  is 
produced  in  a  non-frictional  manner  will  be  represented 
by  the  small  area  PQNM.  Consequently  in  a  finite 

j-B 

non-frictional  operation  AB,  the  heat  absorbed,  I    0d<f>,  is 

represented  by  the  area  of  the  figure  A  Bum,  and  therefore 
generally  depends  on  the  form  of  the  curve  AB  as  well  as 


144  ELEMENTARY   THERMODYNAMICS. 

on  the  initial  and  final  states  A,  B:  in  other  words,  dQ  is 
not  generally  a  complete  differential  of  a  function  of  the 
independent  variables  which  define  the  state  of  the  system. 
The  heat  absorbed  during  the  process  AB  will  be  the 
same  for  all  non-frictional  paths  leading  from  A  to  B,  or 
dQ  a  complete  differential  at  every  point  of  the  path, 
under  any  one  of  the  three  following  conditions  : — 

(1)  When  the  temperature  is  constant  throughout 
every  path ;  or 

(2)  When  the  entropy  is  constant ;  or 

(3)  When  the   temperature  is   a  function   of  the 
entropy. 

64.  The  conception  of  entropy  enables  us  to  remove 
an  objection  which  is  sometimes  urged  against  Carnot's 
principle.  It  is  commonly  stated  that  an  animal  has  a 
much  greater  '  efficiency '  than  a  Carnot's  perfectly  re- 
versible engine  working  through  .the  same  range  of 
temperature,  and  it  is  even  compared  to  an  electro- 
magnetic engine,  although  it  is  obvious  that  there  is  no 
sensible  absorption  of  electric  energy.  In  consequence  it 
is  believed  that  the  changes  which  take  place  in  the 
animal  world  are  not  subject  to  the  restrictions  of  Carnot's 
principle.  The  whole  difficulty,  however,  arises  from  the 
fact  that  the  word  '  efficiency '  is  unconsciously  used  in  a 
new  sense  and  that  it  is  supposed  it  should  then  possess 
the  same  properties  as  when  used  strictly  according  to  the 
definition.  In  Carnot's  principle,  the  '  efficiency '  is  the 
ratio  of  the  work  done  during  a  complete  cycle  to  the 
mechanical  value  of  the  heat  absorbed  from  the  hotter 
body.  In  the  present  example,  the  '  efficiency '  is  the 
ratio  of  the  work  done  in  an  incomplete  cycle  to  the 


CARNOT'S  PRINCIPLE.  145 

mechanical  value  of  the  consequent  change  of  energy. 
If,  for  example,  a  system  undergo  a  reversible  operation, 
at  the  constant  temperature  6,  during  which  the  energy 
and  entropy  change  from  Z7i  and  fa  to  Z72  and  fa,  respec- 
tively, the  heat  given  out  by  the  system  will,  by  equation 
(34)",  be  equal  to  6  (fa  —  $2),  and  therefore  the  work  done 
by  the  system  will  be  U^  -  Uz  -  6  (fa  -  fa).  Thus  the 
ratio  of  the  work  done  to  the  mechanical  value  of  the 
loss  of  energy  is 

*•>,  that  is.  l- 


which,  when  no  sensible  amount  of  heat  is  lost  or  gained, 
will  be  practically  equal  to  unity. 

The  cycle  of  the  animal  body  is  completed  in  the 
vegetable  world,  and  though  very  little  is  really  known 
at  present  of  the  way  in  which  the  growth  of  grass  and 
trees  takes  place,  we  have  little  doubt  that  when  the 
deficiency  is  supplied,  every  part  of  the  cycle  will  be 
found  to  be  in  complete  accordance  with  Carnot's  prin- 
ciple. 

65.  We  have  already  seen  (Art.  25)  that  when  a  body 
is  in  a  state  to  which  the  definition  of  entropy  applies,  the 
velocity  of  its  centre  of  mass  may  be  varied  at  will  by  an 
isentropic  process  without  altering  the  temperature  or  any 
other  property  of  the  body.  Hence  when  two  different 
states  of  the  same  body  or  system  of  bodies  differ  only 
in  the  motion  of  the  centre  of  mass,  the  conception  of 
entropy  will  be  applicable  to  both  states  or  to  neither  ; 
and  in  the  former  case,  the  entropy  will  have  the  same 
value  in  both  states,  so  that  it  is  independent  of  the 
motion  of  the  centre  of  mass. 

P.  10 


146  ELEMENTARY  THERMODYNAMICS. 

If  the  speed  of  rotation  of  a  body  be  altered  by  an 
isentropic  process,  the  temperature  and  other  properties 
of  the  body  will  generally  vary  in  consequence ;  but  if 
the  body  be  made  of  very  rigid  materials,  like  a  fly-wheel, 
these  variations  will  generally  be  extremely  small.  The 
entropy  of  a  rigid  body,  like  a  fly-wheel,  is  therefore  very 
nearly  independent  of  the  speed  at  which  it  is  rotating. 

66.  If  we  take  the  solar  system  for  our  material 
system,  the  conception  of  entropy  will  not  be  applicable, 
on  account  of  the  irreversible  frictional  actions  which  are 
continually  taking  place ;  but  it  will  be  possible,  at  any 
instant,  by  suitable  means,  without  doing  work  on  the 
system,  or  allowing  it  to  receive  or  lose  heat,  to  bring  it 
into  a  slightly  different  state  to  which  the  definition  of 
entropy  applies.  To  do  this,  it  will  be  practically  suf- 
ficient to  stop  the  mechanical  and  thermal  irreversible 
frictional  actions  by  holding  the  tides  rigidly  in  their 
places,  and  preventing  the  conduction  and  radiation  of 
heat  between  parts  of  the  sun's  mass  which  differ  much 
in  temperature  by  enclosing  them  in  good  non-conducting 
materials  able  to  withstand  the  fiercest  temperatures  to 
which  they  may  be  exposed.  Suppose,  then,  that  P  is  the 
actual  state  of  the  solar  system  at  any  instant  and  Q  the 
actual  state  at  any  subsequent  instant,  and  that  P',  Q'  are 
two  other  states  to  which  the  conception  of  entropy 
applies  and  which  differ  very  little  from  P,  Q,  respectively. 
Then  since  the  radiation  of  energy  into  infinite  space  is 
the  only  external  influence  to  which  the  solar  system  is 
subject,  it  is  obvious  that  the  energy  is  less  in  the  state  Q 
than  in  the  state  P ;  and  that  we  may  bring  the  system 
from  the  state  P'  to  the  state  Q'  by  a  reversible  process 


CARNOT'S  PRINCIPLE.  147 

in  which  a  positive  quantity  of  heat  is  lost,  so  that  the 
entropy  in  the  state  Q'  is  less  than  in  the  state  P'.  We 
may  therefore  say  that  the  energy  and  entropy  of  the 
solar  system  are  both  continually  decreasing. 

The  variation  of  the  energy  and  entropy  of  the  solar 
system  is  not  considered  in  the  ordinary  books ;  but  we 
frequently  meet  with  the  discussion  of  a  similar  problem 
relating  to  the  whole  universe  which  seems  to  require 
notice.  In  the  first  place,  the  discussion  in  question 
ignores  the  all-important  distinction  between  bound  ethe- 
rial  energy,  that  is,  the  potential  energy  of  matter,  and 
free  etherial,  or  radiant  energy,  by  classing  the  whole  of 
the  ether  with  material  bodies.  It  is  consequently  con- 
cluded that  the  energy  of  the  universe  is  constant.  In 
the  second  place,  it  is  supposed  that  in  the  equation 


we  may  take  dtdh  to  be  an  element  of  the  radiant  heat 
which  enters  at  the  external  boundary  of  the  ether,  and 
not  an  element  of  the  heat  absorbed  by  some  material 
part  of  the  system,  as  in  the  proof  of  the  formula  given  in 
Art.  62.  With  this  assumption,  every  element  dtdh  is 

zero  and  therefore  the  whole  integral  I  dt  I  -^  is  also  zero, 

so  that  it  would  follow  that  the  entropy  of  the  universe 
continually  increases. 

In  the  present  state  of  our  knowledge,  it  is  perhaps 
unsafe  to  make  any  positive  assertion  as  to  the  energy  or 
entropy  of  the  whole  material  universe. 

67.  We  will  now  consider  the  important  thermo- 
dynamical  problem  of  the  capacity  of  a  material  system 

10—2 


148  ELEMENTARY   THERMODYNAMICS. 

for  doing  mechanical  work  when  it  is  subject  to  certain 
simple  restrictions  as  to  the  way  in  which  it  receives  or 
loses  heat.  The  only  cases  that  will  be  here  investigated 
are  the  following : — 

(1)  The  system  is  supposed  to  be  entirely  prevented 
from   receiving  or  losing  heat,  either  by  conduction  or 
radiation ;  and  the  maximum  amount  of  work  that  can  be 
obtained  from  the  system  under  these  conditions  is  defined 
to  be  the  Adiabatic  Available  Energy. 

(2)  The  system  is  supposed  to  be  prevented  from 
radiating  energy  into  infinite  space  and  from  exchanging 
heat  with  external  bodies  except  with  bodies  A,  B,  ..,, 
which  are  kept  at  any  the  same  constant  uniform  tempe- 
rature 6.     For  this  purpose  it  will  generally  be  necessary 
to    employ  an   intermediate  body  A',  just    as  when  we 
explained  Carnot's  axiom  in  Art.  48.     Under  these  condi- 
tions, the  maximum  amount  of  work  that  can  be  obtained 
from  the  system  is  defined  to  be  the  Available  Energy  at 
the  constant  temperature  6. 

In  both  cases,  the  algebraic  excess  of  the  available 
energy  over  the  mechanical  kinetic  energy  is  defined  to 
be  the  Mechanical  Potential  Energy.  Hence  since  there 
are  two  kinds  of  mechanical  potential  energy,  it  follows 
that  both  kinds  cannot  be  identical  with  true  potential 
energy.  This  result  will  perhaps  be  made  clearer  if  we 
repeat  some  of  our  former  conclusions  relating  to  energy. 

According  to  modern  ideas,  all  actions  at  a  distance 
between  particles  of  matter  are  due  to  continuous  contact 
forces  in  the  ether.  Consequently,  all  energy  is  really 
kinetic  energy,  and  consists  partly  of  the  kinetic  energy 
of  matter  and  partly  of  the  kinetic  energy  of  the  ether. 
The  kinetic  energy  of  matter  may  be  distinguished  as 


CARNOT'S  PRINCIPLE.  149 

mechanical  or  non-mechanical,  kinetic  energy.    The  kinetic 
energy  of  the  ether  may  also  be  divided  into  two  classes : — 

(a)  Etherial  kinetic  energy  free  from  all  material 
restraints,  as  in  waves  of  radiation. 

(6)  Etherial  kinetic  energy  bound  to  material  systems 
and  forming  the  true  potential  energy  of  matter. 
There  is  no  potential  energy  in  the  ether,  because  that 
would  require  actions  at  a  distance  between  the  particles 
of  ether,  which  it  would  need  a  second  kind  of  ether  to 
explain1. 

Now  according  to  the  kinetic  theory  of  gases,  which  is 
not  explained  in  this  book,  the  pressure  of  a  gas  is  due  to 
the  fact  that  its  'molecules'  are  moving  about  freely  among 
one  another  in  all  directions  with  great  velocities.  The 
mechanical  potential  energy  of  a  compressed  gas  therefore 
consists,  to  a  large  extent,  of  the  kinetic  energy  of  matter. 
In  the  case  of  a  bent  spring  it  is  not  so  obvious  at  first 
whether  the  mechanical  potential  energy  consists  of  kinetic 


1  It  may  be  here  pointed  out  that  quite  a  different  meaning  is 
assigned  to  the  term  potential  energy  in  ordinary  dynamics  from  that 
which  is  given  to  it  in  thermodynamics.  In  ordinary  dynamics,  when 
the  expression  for  the  external  work  in  a  small  change  of  state  is  a 
complete  differential,  the  potential  energy  of  the  system  is  defined  to 
be  the  work  which  the  external  forces  can  do  on  the  system  as  it  returns 
from  its  actual  state  to  the  standard  state.  Thus  when  a  stone  is  lifted 
from  the  ground,  it  is  usually  said  to  acquire  potential  energy.  In 
thermodynamics,  if  the  stone  be  considered  as  a  complete  system  in 
itself,  it  is  said  to  acquire  no  potential  energy ;  but  if  we  suppose  it 
to  form  part  of  one  system  with  the  earth,  then  the  system  acquires 
additional  mutual  potential  energy.  But  we  cannot  assert  that  this 
mutual  potential  energy  belongs  necessarily  to  the  stone ;  for  if  the  stone 
fall  to  the  earth,  the  earth  rises  to  meet  the  stone,  so  that  the  kinetic 
energy,  which  takes  the  place  of  the  mutual  potential  energy,  is  divided 
between  them. 


150  ELEMENTARY   THERMODYNAMICS. 

or  potential,  energy ;  that  is,  whether  the  mechanical  po- 
tential energy  may  not  be  due  to  actions  at  a  distance. 
But  even  in  this  case,  there  is  little  doubt  it  will  be 
found  to  consist,  to  a  large  extent,  of  the  kinetic  energy 
of  matter.  In  the  vibrations  of  a  spring  we  therefore  see 
the  transformation  of  non-mechanical  into  mechanical, 
kinetic  energy,  and  conversely. 

68.  When  a  material  system  has  been  deprived  of  its 
available  energy,  it  is  evident  that  its  mechanical  motions, 
both  of  translation  and  rotation,  will  be  zero,  and  that  the 
temperatures  of  all  its  parts  will  be  equal.  The  ultimate 
electric  and  magnetic  condition  of  the  system  will  not  be 
discussed  here,  but  may  be  understood  from  works  on 
electricity.  As  to  the  relative  positions  of  its  parts,  it  is 
obvious  that  when  any  portion  of  the  system  is  gaseous, 
it  may  be  necessary  to  impose  some  minor  restrictions  in 
addition  to  those  which  specify  that  the  system  is  not  to 
receive  or  lose  heat,  or  only  to  exchange  heat  with  bodies 
A,  B,  ...,  which  are  kept  at  the  same  constant  uniform 
temperature.  For  instance,  if  the  gas  be  contained  in  a 
closed  vessel,  we  may  impose  the  condition  that  the  vessel 
is  to  be  kept  gas-tight ;  if  the  vessel  be  in  the  form  of  a 
cylinder  fitted  with  a  gas-tight  piston,  we  may  suppose 
that  the  expansion  is  not  to  proceed  beyond  a  certain 
point,  or  that  the  final  pressure  is  to  have  a  given  value. 
Even  when  the  system  is  entirely  solid  or  liquid,  we  may 
have  minor  restrictions ;  for  it  may  be  required  that  all 
parts  of  the  system  preserve  their  individual  identity,  or 
we  may  be  at  liberty  to  weld  them  all  into  one  homo- 
geneous mass.  The  minor  restrictions,  it  may  be  remarked, 
are  generally  understood  without  being  expressly  stated, 


CARNOT'S  PRINCIPLE.  151 

and  are  not  to  prevent  the  ultimate  exchange  of  heat 
between  different  parts  of  the  system  or  between  the 
system  and  the  bodies  A,  B, 

69.  In  order  to  apply  the  principle  of  reversibility  to 
the  investigation  of  available  energy,  it  is  necessary  to 
restrict  ourselves  to  those  states  of  the  system  to  which 
the  conception  of  entropy  applies. 

In  the  first  place,  let  the  system  be  prevented  from 
receiving  or  losing  heat,  and  suppose  that  it  is  brought 
by  a  reversible  path  from  an  initial  state  P,  to  which  the 
conception  of  entropy  is  applicable,  to  a  final  state  Z,  in 
which  the  available  energy  is  zero.  Then  the  entropy 
is  the  same  in  the  state  Z  as  in  the  state  P ;  and  it  may 
be  assumed,  as  a  result  of  experience,  that  if  the  system 
be  subject  to  the  same  minor  restrictions  as  before,  and 
be  brought  by  a  different  reversible  path  from  the  initial 
state  P  to  a  final  state  F,  the  states  F  and  Z  will  be 
identical.  Let  these  paths  be  represented  by  (1)  and  (2), 
and  denote  the  corresponding  amounts  of  work  obtained 
from  the  system  by  W±  and  TF2.  Then  if  the  system  be 
made  to  pass  from  P  to  Z  by  (1),  and  then  to  return  from 
Z  to  P  by  (2),  it  will  have  undergone  a  complete  rever- 
sible cycle  in  which  there  is  neither  loss  nor  gain  of  heat. 
The  work  TFj  —  TF2  obtained  from  the  system  must  there- 
fore be  zero,  or  Wl  =  W2, 

If  the  system  be  brought  from  P  by  a  frictional  path 
to  a  final  state  Z',  in  which  the  available  energy  is  zero, 
the  state  Z'  cannot  be  the  same  as  Z;  because  if  <f>  be  the 
entropy  in  the  state  P,  the  entropy  in  the  state  Z  will  be 
equal  to  <f>,  and  in  the  state  Z',  greater  than  <£.  The  system, 
however,  will  evidently  admit,  in  general,  of  being  brought 


152  ELEMENTARY    THERMODYNAMICS. 

from  Z  to  Z'  by  a  frictional  path  in  which  a  positive 
quantity  of  work  is  expended  (in  friction).  Suppose,  then, 
that  W  is  the  work  obtained  from  the  frictional  path  PZ', 
and  W  the  work  obtained  from  any  reversible  path  PZ, 
and  let  the  system  be  brought  from  P  to  Z'  in  the  two 
following  ways : — 

(1)  Let  the  system  travel  by  the  frictional  path  PZ' 
yielding  a  quantity  of  work  W. 

(2)  Let  the  system  first  travel  from  P  to  Z  by  a 
reversible  path,  giving  out  a  quantity  of  work   W;   and 
then  from  Z  to  Z ',  giving  a  negative  quantity  of  work, 
—  x  say,  where  x  is  positive.     Then  since  there  is  neither 
loss  nor  gain  of  heat  in  the  paths  leading  from  P  to  Z', 
the  principle  of  energy  gives 

W'=W-x, 
or  W>W. 

Hence  if  U  and  U0  be  the  values  of  the  energy  in  the 
states  P  and  Z,  the  adiabatic  available  energy  in  the  state 
P  is  U—U0,  with  the  condition  </>  =  <£0. 

Secondly,  let  us  suppose  that  the  system  exchanges 
heat  with  bodies  which  are  kept  at  the  same  constant 
uniform  temperature  0.  Then  if  the  system,  subject  to 
given  minor  restrictions,  be  brought  by  any  operation 
from  the  state  P  to  a  final  state  Z,  in  which  the  available 
energy  is  zero,  it  is  obvious  that  the  temperature  of  the 
system  will  then  be  uniform  and  equal  to  6,  and  that  the 
state  Z  will  be  the  same  for  all  operations.  From  this  it 
may  be  shown  that  the  work  W  obtained  from  the  system 
as  it  travels  from  P  to  Z  is  the  same  for  all  non-frictional 
paths,  and  is  then  greater  than  for  any  frictional  path. 
Since,  in  a  non-frictional  path,  no  part  of  the  system  is 
allowed  to  receive  or  give  out  heat  except  when  its  tern- 


CARNOT'S  PRINCIPLE.  153 

perature  is  6,  if  (  U,  <J>),  (  Uz,  <f>z)  be  the  energy  and  entropy 
of  the  system  in  the  states  P  and  Z,  respectively,  the  heat 
absorbed  in  a  reversible  path  leading  from  P  to  Z  is 
6  (<f>z  —  $),  or  the  heat  given  out  is  6  (tf>  —  </>z),  and  therefore 
the  work  obtained  from  the  path  is  U  —  Uz  —  6  (<f>  —  <j>z). 
Writing  f  for  U—6<f>,  this  becomes  7  —  fz,  where,  it 
must  be  remembered,  <f>  is  not  equal  to  <f>2. 

70.  If,  while  a  system  is  protected  from  all  external 
influences,  including  as  such  the  radiation  of  energy  into 
external  space,  an  action  takes  place  within  the  system 
by  which  the  state  is  changed  from  P  to  P',  the  energy 
will  remain  the  same  but  the  entropy  will  be  increased. 
Thus  if  (U,  <£),  (IT,  <f>')  be  the  energy  and  entropy  in  the 
states  P,  P',  respectively,  we  shall  have 
U'=U] 


Now  let  W,  W,  be  the  adiabatic  available  energies  in 
the  states  P  and  P',  Z  and  Z'  the  corresponding  final 
states;  and  let  U0,  U0',  be  the  values  of  U  in  the  final 
states  Z,  Z'.  Then  we  have 

W=U-U0 


so  that  TF-F'=[7o'-Z7o. 

But  since  the  entropy  in  the  final  state  Z'  is  equal  to  </>', 
and  in  the  final  state  Z  equal  to  <£,  which  is  less  than  <f>', 
and  since  there  are  no  mechanical  motions  in  either  state, 
it  may  be  inferred  that  the  system  can  be  brought  from  Z 
to  Z'  by  a  frictional  path  in  which  a.  positive  quantity  of 
work  is  expended  (in  friction)  and  no  heat  either  gained 
or  lost.  We  therefore  have  Ut)'  >  U0,  so  that  W>  W. 
Again,  if  V,  V,  be  the  available  energies  at  the  con- 


154  ELEMENTARY   THERMODYNAMICS. 

stant  temperature  0  in  the  two  states  P,  P',  and  if  Z  be 
the  common  final  state, 

F=y  -5 


so  that 


and  therefore  V>V. 

Since,  therefore,  when  a  system  is  left  to  itself,  the 
entropy  has  a  constant  tendency  to  increase,  it  follows 
that  there  is  a  constant  tendency  to  a  loss  of  available 
energy  ;  in  other  words,  to  a  transformation  of  available 
into  unavailable,  energy.  This  result  is  the  great  prin- 
ciple of  the  Degradation  of  energy,  first  enuntiated  by 
Sir  W.  Thomson.  It  was  formerly  known  as  the  principle 
of  the  Dissipation  of  energy  ;  but  the  use  of  the  word 
dissipation  is  now  generally  abandoned,  since,  as  we  have 
seen,  there  is  no  actual  loss  of  energy  when  the  system 
passes  from  P  to  P'. 

Again,  since  it  is  obvious  that  when  a  finite  system  is 
left  to  itself,  the  entropy  cannot  increase  indefinitely,  we 
conclude  that  the  system  will  ultimately  settle  down  into 
a  permanent  state  or  condition,  as  stated  in  Art.  26.  The 
principal  agency  by  which  this  condition  is  generally 
brought  about  will  be  explained  a  little  later  on  ;  in  the 
meantime  it  may  be  stated  that  the  ultimate  condition  is 
not  necessarily  such  as  to  make  the  available  energy  zero. 

71.  When  we  wish  to  obtain  the  maximum  amount 
of  work  from  a  system  which  is  prevented  from  radiating 
energy  into  infinite  space,  and  also  from  exchanging  heat 
with  any  other  system,  or  which  is  only  allowed  to  ex- 


CARNOT'S  PRINCIPLE.  155 

change  heat  with  bodies  which  are  kept  at  the  same 
constant  uniform  temperature,  we  are  at  liberty  to  bring 
the  system  to  its  final  state  by  any  reversible  process  we 
please.  If,  for  example,  the  system  consists  of  a  number 
of  different  bodies  or  parts,  we  may  begin  by  depriving 
each  part  of  its  available  energy.  If,  during  the  rest  of 
the  process,  the  state  of  each  part  can  be  kept  invariable, 
we  shall  have  Prof.  Tait's  proposition  that  the  available 
energy  of  a  system  is  equal  to  the  sum  of  the  available 
energies  of  each  of  its  parts  together  with  their  mutual 
available  energy  after  their  individual  available  energies 
have  been  exhausted. 

We  will  now  take  a  few  particular  examples  of  adia- 
batic  available  energy  for  systems  in  which  there  are  no 
manifestations  of  electricity  or  magnetism. 

(1)  Let  the  system  consist  of  a  number  of  bodies  A 
together  with  a  vast  body  B  whose  temperature  is  uniform 
and  equal  to  00  and  whose  volume  is  constant.  Then  since 
the  maximum  amount  of  work  may  be  obtained  by  any 
reversible  path  we  please,  we  will  take  the  simplest,  as 
follows.  Let  the  bodies  A  undergo  any  set  of  isentropic 
operations  by  which  the  temperature  of  every  part  is 
reduced  to  #0,  and  then  let  them  be  put  in  thermal  com- 
munication with  B  and  bring  the  system  to  its  final  state 
by  any  set  of  isothermal  operations.  If  (U,  $)  be  the 
original  values  and  (U0,  <£0)  the  final  values,  of  the  energy 
and  entropy  of  the  bodies  A,  the  heat  given  out  by  them 
to  B  will,  by  equation  (34-)",  be  equal  to  00  (<f>  —  </>0),  and 
therefore  the  maximum  amount  of  work  which  can  be 
obtained  from  the  system,  or  its  adiabatic  available  energy, 
will  be 

t/r-tf0-00(0-00) (37). 


156  ELEMENTARY   THERMODYNAMICS. 

Putting  y  for  U—0^,  this  becomes  7  -  f  0. 

In  this  example  we  have  evidently  found  the  available 
energy  at  the  constant  temperature  #0,  of  the  bodies  A 
alone. 

(2)  Let  the  system  consist  of  masses  ml)  m.2,  m3,..,mn 
of  the  same  kind  of  perfect  gas  contained  in  cylinders 
fitted  with  air-tight  pistons  whose  volumes  may  be  varied 
in  any  way  subject  to  the  condition  that  the  total  volume 
remains  constant.  Also  let  the  temperatures  of  the  diffe- 
rent masses  be  originally  6l,6.2,6^,...0n,  and  their  volumes 
per  unit  mass  vl}  v.>,  vs,  ...  vn,  respectively.  Then  since  for 
a  unit  mass  of  perfect  gas  we  have  very  approximately 
dU—Cv  dd,  and  since  the  work  done  on  the  gas  in  a  small 
reversible  operation  is  —  pdv,  the  equation  d  U  —  dQ  +  dW 
gives  for  a  small  reversible  process 


or  since  the  gas  satisfies  very  approximately  the  relation 


Hence  since  -~  =  d<f>, 

we  have  d<t>  =  C^+Rdv. 

u  V 

Integrating  and  remembering  that  Ce  is  practically 
stant,  we  obtain 


Thus  when  the  given  masses  of  gas  are  brought  in  any 
reversible  way  to  a  final  state  in  which  each  quantity  has 
the  same  pressure  p0  and  the  same  temperature  00,  the 


CARNOT'S  PRINCIPLE.  157 

conditions  that  the   total  entropy  and  the  total   volume 
are  the  same  as  at  first,  are 

J*  , 


Cv 

0, 


and 
Hence 


x  (v™  vam>  ......  vnm")    ...............  (38), 

which  gives  B0.     To  find  p0,  we  use  the  equation 


The  equation  dU=Cvdd,  or  U- U'  =  Cv(d-6'\  then 
shows  that  the  work  obtained  from  the  system  is 

If  the  masses  m1}  m*,  ms, mn,  are  each  equal  to  m, 

equation  (38)  becomes 

Cr  C,  1 

— —      nfii 
V 

The  foregoing  examples  are  taken  substantially  from 
Maxwell's  Theory  of  Heat.  The  following  is  due  to  Sir 
W.  Thomson,  but  has  been  somewhat  modified  by  Prof. 
Tait. 

(3)  Let  there  be  two  equal  homogeneous  bodies  of 
masses  m,  of  the  same  kind  of  substance,  without 
mechanical  kinetic  energy  but  at  different  uniform  tern- 


158  ELEMENTARY   THERMODYNAMICS. 

peratures  01}  0.2,  and  suppose  that  the  alterations  of  shape 
and  volume  can  be  neglected.  Then  no  work  can  be 
obtained  from  the  system,  and  we  shall  therefore  suppose 
that  we  are  at  liberty  to  equalize  the  temperatures  of  the 
two  bodies  by  means  of  a  Carnot's  perfectly  reversible 
engine  which  performs  complete  cycles.  Also,  for  sim- 
plicity, we  shall  take  the  specific  heat  to  be  independent 
of  the  temperature ;  and  in  making  the  calculations,  the 
Carnot's  engine  may  be  supposed  to  work  between  the 
two  given  bodies  and  any  third  body  whose  temperature 
is  kept  at  a  uniform  constant  value  00,  provided  that,  on 
the  whole,  no  heat  is  thus  imparted  to,  or  abstracted  from, 
the  third  body.  The  temperature  00  will  evidently  be  the 
final  temperature  of  the  two  given  bodies. 

If,  then,  the  temperature  of  one  of  the  given  masses 
be  increased  by  d0,  the  heat  imparted  to  it  will  be  mcd0, 
and  if  dq0  be  the  corresponding  quantity  of  heat  taken 
from  the  third  body,  we  have,  by  the  definition  given  in 
equation  (29), 

mcdd  _  dq0 

Thus  the  condition  that,  on  the  whole,  heat  is  neither 
gained  nor  lost  by  the  third  body,  gives 

v*de 


that  is>  log    ?  +  log  £  =  0  ...............  (40), 

or  0i  =  0A, 

whence  we  find  the  final  temperature, 


(41). 


CARNOT'S  PRINCIPLE.  159 

The  same  result  might  have  been  obtained  from  the  con- 
sideration that  the  entropy  remains  constant. 

Again,  in  the  cycle  in  which  the  engine  takes  a 
quantity  of  heat  dq0  from  the  third  body  and  gives  a 
quantity  mcdd  to  the  other,  the  work  done  on  the  engine 
is  evidently 

mcdd  —  dq0, 

or  mcdd  (l-\. 


The  maximum  amount  of  work,  W,  which  can  be  obtained 
from  the  two  bodies  by  means  of  the  Carnot's  engine  will 
therefore  be 


that  is,  by  equation  (40), 


=  »ic(V0i-<W  ..................  (42). 

More  generally,  if  there  be  n  equal  homogeneous  bodies 
of  the  same  kind  of  matter,  without  mechanical  kinetic 
energy  but  at  different  uniform  temperatures;  then  if 
the  specific  heat  be  independent  of  the  temperature, 
we  shall  have 


that  is, 


or 

and  therefore  0Q  =  \/W^A.  .....  0n  ...............  (43). 


160  ELEMENTARY    THERMODYNAMICS. 

Also  W=  me  (  I'1  d0+l'*d0  + 

\Je0  Je0 


The  adiabatic  available  energy  of  the  bodies  is  therefore 
proportional  to  the  excess  of  the  Arithmetic  mean  over 
the  Geometric  mean,  of  their  absolute  temperatures. 

More    generally   still,    if    there    be   any    number    of 
unequal   bodies,  without    mechanical  kinetic  energy,  the 
temperature  of  each  being  uniform  and  the  specific  heats 
independent  of  the  temperature,  we  shall  have 
d0  dd 


that  is,        WA  log      +  m.$.2  log  -    +  ......  =0, 


™*"' 


W=  mfr  I  d0+  m.cn  \d0+  ...... 

J  8»  J  0» 


or 


And 


72.  The  results  which  have  now  been  given  in  this 
and  in  the  first  chapter  will  be  more  completely  explained 
by  considering  some  of  the  chief  mechanical  and  thermo- 
dynamical  questions  arising  out  of  the  problem  of  the 
solar  system.  This  problem  is  based  on  the  following 
empirical  laws,  known  as  the  Three  Laws  of  Kepler  :  — 

I.  The  planets  describe  elliptic  orbits  about  the  sun, 
which  remains  fixed  in  one  focus. 

II.  As  any  planet  moves  in  its  orbit,  the  straight 


CARNOT'S  PRINCIPLE.  161 

line  which  joins  it  to  the  sun  describes  equal  areas 

in  equal  times. 

III.     The  squares  of  the  periodic  times  of  any  two 

planets   are   to   one   another  as   the   cubes  of    their 

mean  distances  from  the  sun. 

These  laws,  published  in  1619,  were  deduced  by  Kepler 
by  an  immense  amount  of  arithmetical  labour  from  the 
observations  of  Tycho  Brahe.  They  are  not  quite  exact, 
as  we  shall  see  presently.  Hence,  as  the  magnitudes  of 
the  sun  and  the  planets  are  small  compared  with  their 
distances  from  one  another,  it  will  be  sufficient  to  begin 
by  treating  them  as  mere  particles. 

73.  Kepler's  laws  lead  immediately  to  Newton's  great 
principle  of  universal  gravitation,  first  published  in  1687. 
The  steps  of  the  argument  may  be  briefly  presented  as 
follows : 

(a)  Since  no  planet  moves  with  uniform  velocity  in  a 
straight  line,  it  follows  that  every  planet  is  acted  on  by  a 
force.  Also  since  the  orbit  of  every  planet  is  plane,  the 
acceleration  at  right  angles  to  the  plane  of  the  orbit  is 
constantly  zero.  The  resultant  force  on  the  planet  there- 
fore acts  in  the  plane  of  the  orbit. 

(6)     If  8  be  the  position  of  the  sun,  P,  Q  the  positions 


of  a  planet  at  any  two  consecutive  instants,  then  if  p  be 

the  perpendicular  from  S  on  the  tangent  at  P  and  the 

P.  11 


162  ELEMENTARY   THERMODYNAMICS. 

elementary  arc  PQ  be  denoted  by  ds,  the  elementary  area 
PSQ  will  be  equal  to  %pds. 

Hence  if  -,    be  the  rate  of  description  of  area  by  the  line 
dt 

SP,  we  have 

Pdi  =  2  ~dt' 

or,  if  v  be  the  velocity  of  the  planet  in  its  orbit, 
~dA 


Multiplying  both  sides  by  P,  the  mass  of  the  planet  in 
grammes,  we  get 


Now  the  left-hand  side  is  the  planet's  moment  of  momen- 
tum about  a  straight  line  through  S  at  right  angles  to  the 
orbit,  and  by  Kepler's  second  law,  the  right-hand  side  is 
constant.  But  we  have  already  seen  that  the  rate  at 
which  the  angular  momentum  of  a  particle  about  any 
fixed  line  increases  with  the  time  is  equal  to  the  moment 
of  the  resultant  force  which  acts  upon  it.  The  moment 
of  the  resultant  force  in  our  case  is  therefore  zero.  Con- 
sequently, the  force  which  acts  on  the  planet  tends  to  or 
from  S.  Since,  by  Kepler's  first  law,  the  orbit  is  always 
concave  to  8,  it  is  easily  seen  that  the  force  always  acts 
from  P  to  S  ;  in  other  words,  that  the  planet  is  always 
attracted  by  the  sun. 

(c)  If  F  be  the  resultant  force  in  dynes  which  acts 
on  the  planet  P,  considered  positive  when  it  tends  to  the 
sun,  and  if  SP,  the  distance  of  the  planet  from  the  sun, 
be  denoted  by  r,  the  work  done  on  the  planet  in  a  small 
displacement  will  evidently  be  —  Fdr  ergs.  Hence,  if 


CARNOTS  PRINCIPLE. 


163 


(v,  r),  (v0,  r0)  be  the  values  of  v  and  r  at  any  two  instants, 
the  principle  of  work  gives 

-v02)  =  -r  Fdr, 


the  integral  sign  merely  denoting  a  summation,  without 
implying  that  F  is  a  function  of  r  only. 
Substituting  for  v  from  equation  (45),  we  obtain 


2P   ; 


]L_  i\  =  _  r 

p0V  Jro 


Fdr 


.(46). 


Now  if  (p'}  r)  denote  quantities  corresponding  to  (p,  r) 
with  respect  to  the  other  focus  of  the  elliptic  orbit,  and 


Pf   \ 


if  2a,  26  be  the  principal  axes,  we  have  the  following 
well-known  properties : — 

r  +  r'  =  2a, 

p     £ 

r     r' 

Multiplying  each  side  of  the  last  of  these  results  by  p, 
we  easily  find 


=        = 
r      r'      2a-r 


that  is 


11—2 


164  ELEMENTARY   THERMODYNAMICS. 

Hence  by  equation  (46), 


Differentiating  with  respect  to  r,  and  remembering  that 

-j—  is  constant,  we  find 
at 


which  may  be  more  simply  written 


where  u  denotes  the  constant  factor  (  -=-  )  -=• 

\dtj   b- 

In  like  manner,  the  attractive  force  F',  exerted  by  the 
sun  on  any  other  planet  of  mass  P'  at  a  distance  r',  may 
be  shown  to  be 


( 
have 

^ 

T= 

dt 


(d)     If  T  be  the  periodic  time  of  the  planet  P,  we 
e 


^       TTO 

T=dA 


and  therefore        T*  =  7ra   a  =       a* . 

(dt) 
Similarly,  2"2  =    7ra ~. 

Hence,  a  being  the  average   of  the   greatest  and  least 
distances  of  the  planet  P  from  the  sun,  or  what  is  usually 


CABNOT'S  PRINCIPLE.  165 

called  its  mean  distance,  Kepler's  third  law  gives  /*  =  /&'. 
We  therefore  have  F=  ^—  and  F  =  ^~  ,  so  that  the  sum 

attracts  the  different  planets  with  forces  proportional  to 
their  masses  and  inversely  proportional  to  the  square  of 
their  distances. 

(e)  Again,  if  we  consider  only  motion  relative  to  the 
earth,  the  orbit  of  the  moon  is  found  to  be  approximately 
an  ellipse  described  so  that  the  line  which  joins  the  moon 
to  the  earth  sweeps  over  equal  areas  in  equal  times.  We 
therefore  infer  that  the  moon  is  attracted  to  the  earth  by 

A  force     —  ,  where  M  is  the   mass   of  the   moon,  r   its 

distance  from  the  earth,  and  v  a  constant  which  is  not  to 
be  assumed  equal  to  p.  Now  if,  as  a  rough  approximation, 
we  were  to  take  the  moon's  orbit  about  the  earth  to  be 
circular,  the  force  by  which  the  moon  is  attracted  to  the 
earth  would  always  be  at  right  angles  to  the  direction  of 
motion,  and  therefore  the  velocity  would  be  constant. 
Hence  if  we  denote  the  angular  velocity  of  the  moon 
about  the  earth  by  &>,  o>  would  be  constant,  and  the 
resultant  acceleration  would  tend  to  the  centre  of  the 
orbit  and  be  equal  to  rar.  Substituting  for  r  and  <o  their 
known  values,  we  find  (in  the  c.  G.  s.  units) 


27x24x  (50x60 

=  '27, 


so  that  -  =  -27. 

r- 


But  we  know  that  gravity,  that  is,  the  force  by  which  the 
weight  of  bodies  is  caused,  is  proportional  to  the  mass 


166  ELEMENTARY  THERMODYNAMICS. 

acted  on,  and  at  any  point  of  the  earth's  surface,  the  force 
of  gravity  on  a  mass  of  one  gramme  is  about  980  dynes. 
Hence,  since  the  mean  distance  of  the  moon  from  the 
earth  is  about  60  times  the  earth's  radius,  if  the  force  of 
gravity  decrease  inversely  as  the  square  of  the  distance, 
the  acceleration  towards  the  earth  caused  by  gravity  in 
any  body  at  the  distance  of  the  moon,  would  be  about 


From  this  we  draw  the  important  conclusion  that  the 
force  by  which  the  moon  is  caused  to  revolve  about  the 
earth  is  identical  with  terrestrial  gravity. 

(/)  To  effect  a  rough  comparison  between  /u,  and  v, 
it  will  be  sufficient  to  treat  the  orbit  of  the  earth  about 
the  sun  as  circular.  Taking  (r,  co)  to  refer  to  the  moon's 
orbit  about  the  earth,  and  (R,  H)  to  the  earth's  orbit 
about  the  sun,  we  have 


v  =  r^w-  }  ' 
But,  roughly, 

R  =  400r) 


-« 

hence  fj,  is  much  greater  than  v. 

(g)  From  the  preceding  results  we  infer  Newton's 
great  principle  of  universal  gravitation,  which  asserts  that 
every  portion  of  matter  in  the  universe  attracts  every 
other  portion  according  to  the  following  simple  law  :— 


CARNOT'S  PRINCIPLE.  167 

Let  m  and  m'  be  the  masses  of  any  two  portions  of  matter 
and  r  their  distance  apart,  the  dimensions  of  m  and  m' 
being  small  compared  with  r  ;  then  m  and  m'  exert  equal 

and  opposite  attractions  of  X      —  dynes  on  one  another, 

where  X  is  a  constant  number  which  is  independent  of 
the  natures  of  m  and  m',  and  of  their  chemical  and  physical 
states,  and  also  of  the  presence  or  absence  of  other  matter. 
The  rigid  accuracy  of  the  law  of  gravitation  is  proved  by 
the  fact  that  a  vast  number  of  intricate  calculations  have 
been  performed  in  which  it  has  not  once  been  found  to  lead 
us  wrong.  The  masses  m  and  m  may,  if  we  please,  be  as 
small  as  possible,  that  is,  in  chemical  language,  m  and  m' 
may  be  the  masses  of  atoms.  We  cannot,  however,  assert 
that  the  force  of  gravitation  exists  between  the  different 
particles  of  the  same  atom. 

The  universal  prevalence  of  the  attraction  of  gravita- 
tion has  an  important  bearing  upon  thermodynamics, 
because  it  will  generally  produce  or  modify  motion,  and 
motion  is  generally  attended  by  friction. 

The  fact  that  the  values  of  /*  and  v  are  different  is 
easily  explained  by  the  law  of  gravitation.  Thus  if,  for  sim- 
plicity, we  treat  the  sun,  earth  and  moon,  as  mere  particles, 
and  if  we  denote  their  masses  by  S,  E,  M,  the  attractions 
between  the  sun  and  earth,  and  between  the  earth  and 

«  r>  Tf  M 

moon,  will  be  respectively  equal  to  \  ^  and  X,  -^  .     But 

these  forces  have  been  denoted  by  ^  and  v—  .  Hence 
/m  =  \S,  and  v  =  \E,  so  that 


168  ELEMENTARY   THERMODYNAMICS. 

Since,  according  to  Newton's  law  of  gravitation,  the 
planets  attract  the  sun  as  much  as  the  sun  the  planets,  it 
is  clear  that  the  sun  cannot  remain  immovable,  as  Kepler 
supposed  ;  but  owing  to  his  great  mass,  his  motion  will  be 
very  small.  Again,  since  the  planets  are  not  only  attracted 
by  the  sun  but  by  one  another,  it  is  evident  that  their 
orbits  cannot  be  perfect  ellipses.  These  questions  will 
be  found  considered  in  detail  in  works  on  the  lunar  and 
planetary  theories. 

Some  rough  approaches  to  the  law  of  gravitation  had 
been  made  before  the  appearance  of  Newton's  great  dis- 
coveries, by  considering  the  case  of  an  orbit  perfectly 
circular;  but  it  seems  that  no  one,  with  the  exception, 
perhaps,  of  Horrox,  had  supposed  that  the  forces  by  which 
the  moon  and  planets  are  compelled  to  describe  their 
orbits,  are  the  very  forces  which  produce  terrestrial 
gravity.  To  show  the  connection  of  circular  motion  with 
Kepler's  laws,  let  a  circular  orbit  of  radius  r  be  described 
with  uniform  angular  velocity  o>  by  a  body  of  mass  M. 
Then  the  body  will  be  acted  on  by  a  constant  force  Mra>- 
tending  to  the  centre,  and  therefore  if  the  central  force  be 

-^-  ,  we  shall  have 

fi  =  r'to2. 
Now  if  T  be  the  periodic  time,  we  have 


so  that  T2=- 

P* 

Similarly,  if  a  second  body  of  mass  M'  describe  another 


CARNOT'S  PRINCIPLE.  169 

/  Tr07 

circular  orbit,  of  radius  r',  under  a  central  force  —  ^-  ,  the 
periodic  time  T  will  be  given  by 


A* 

Hence  T-  :  T*  =  r*  :  r'3, 

which  is  Kepler's  third  law. 

74.  If  a  particle  P  be  situated  so  near  a  finite  body 
B  that  it  cannot  be  supposed  at  the  same  distance  from 
every  part  of  it,  it  will  be  necessary,  in  order  to  find  the 
resultant  attraction  of  B  on  the  particle  P,  to  imagine  the 
body  B  divided  into  a  large  number  of  parts,  each  of  which 
is  so  small  in  all  its  dimensions  that,  for  the  purpose  of 
finding  its  attraction  on  P,  it  will  be  sufficient  to  treat  it 
as  a  mere  particle.  The  simplest  and  most  useful  case  to 
consider  is  that  in  which  B  is  in  the  form  of  a  sphere, 
either  of  uniform  density  throughout,  or  such  that  the 
strata  of  uniform  density  are  shells  bounded  by  spheres 
concentric  with  B. 
Let  the  figure  represent  such  a  shell,  of  density  p  and 


Q' 

indefinitely  small  thickness  T.     Join  P  to  0,  the  centre  of 
the  shell,  and  take  a  point  C  in  OP  such  that 

0(7.  OP  =  a2, 
where  a  is  the  radius  of  the  shell.     With  C  as  vertex 


170  ELEMENTARY   THERMODYNAMICS. 

and  any  small  vertical  solid  angle  dw,  describe  any  cone 
cutting  out  a  small  area  da-  from  the  sphere  at  Q  and 
another  small  area  da'  at  Q1.  Then  since  OQ  and  OQ' 
make  the  same  angle,  0  say,  with  the  line  QCQ',  the 
tangent  planes  to  the  sphere  at  Q  and  Q'  will  each  be 
inclined  at  the  angle  6  to  any  plane  normal  to  QCQ'. 
Hence 

da-  cos  0  =  <7Q2  .  dm] 

da-'  cos  6  =  CQ'2 .  dw}  ' 

Again,  if  ra  be  the  mass  of  the  particle  P,  the  attractions 
of  the  elements  at  Q  and  Q'  on  P  will  be  respectively 

mprda-       ,      mprda-' 

equal  to  X—^,^    and  X    7\pD^~ '   Substituting  for  acr  and 
V*  (^Jr- 

j  ,  ,,  ,  mprdd)   CO2       , 

da- ,  these  expressions  become  \          (T  •  rjp2  an^ 

mprdto    CQ'2 
cos  (9  "'QrP*' 

Now  if  we  join  the  lines  as  in  the  figure,  we  have 
OC  _OQ 
OQ~OP' 

Q 


The  angle  OPQ  is  therefore  equal  to  OQG,  that  is,  to  6. 
In  like  manner,  the  angle  OPQ'  is  equal  to  0. 
Also,  by  similar  triangles,  we  have 

OQ_OP 

QC'PQ' 


CARNOT'S  PRINCIPLE.  171 


CQ'2       a2 
Similarly,  Q^=OP-' 

Thus  the  attractions  of  da-  and  da-'  on  P  are  each  equal  to 
X  —  —  -£-  .  ,jp2  .     They  both  lie  in  a  plane  through  OP, 

and  make  the  same  angle  0  with  OP.     Their  resultant 
therefore  acts  along  PO  and  is  equal  to 


Hence  also  the  resultant  attraction  of  the  whole  shell  on 
P  acts  along  PO  and  is  equal  to  X  ^Wp^  »  tna*  ^»  ^ 

is  the  same  as  if  the  whole  shell  were  condensed  into  a 
particle  at  0.  This  result,  being  true  for  every  such  shell, 
is  true  for  the  whole  sphere ;  so  that  if  M  be  the  mass  of 
the  sphere  and  r  the  distance  OP,  the  attraction  of  the 
sphere  on  P  will  be 

Mm  l 

A*       ~    . 

r8 

Now  let  0'  be  the  centre  of  another  sphere  B',  which, 
like  B,  is  either  of  uniform  density  throughout,  or  such 
that  the  strata  of  uniform  density  are  shells  bounded  by 
spheres  which  have  a  common  centre  at  0'.  Then  in 
order  to  find  the  attraction  of  B  on  B',  we  shall  first  find 
the  attraction  of  B  on  one  of  the  thin  uniform  shells  of 
which  B  is  composed.  Let  a  be  the  radius  of  such  a 

1  The  foregoing  proof  is  from  Thomson  and  Tait,  by  whom  it  has 
been  modified  from  Newton. 


172  ELEMENTARY   THERMODYNAMICS. 

shell,  r'  its  indefinitely  small  thickness,  and  p  its  density. 
Join  00'  and  take  a  point  (7  in  00'  such  that 

O'C' .  O'O  =  a'-\ 

With  6"  as  vertex  and  any  small  vertical  solid  angle  dot', 
describe  any  cone   cutting  out  small   portions  from   the 


spherical  shell  0',  at  q  and  q'.  Then  it  may  be  shown 
that  the  four  angles  O'qq',  O'q'q,  qOO',  q'OO',  are  equal  to 
one  another,  and  we  may  therefore  denote  each  of  them 

by   6'.     It    may  also    be   shown   that    —%.;  —    'n°  =  TTD'2 ' 

Again,  in  order  to  find  the  attraction  of  the  sphere  B  on 
the  elements   at   q   and  q,  we  may  suppose  the  sphere 
concentrated  into  a  point  at  0,  and  the  elements  at  q  and 
q  may  be  treated  as  particles.     It  will  thence  follow  that, 
if  M  be  the  mass  of  the  sphere  B,  the  attractions  of  B 
on  the  elements  at  q  and  q'  are  each  equal  to 
Mp'r'dw      a'2 
'    cos0'     'OO72' 

and  act  along  qO  and  q'O,  respectively. 
These  two  forces  may  be  compounded  in  the  usual  way. 

Thus,  let  two  forces,  each  equal  to  X      ^/v"  •  T^TY-..  ^e 

applied  at  0'  in  opposite  directions  parallel  to  qO.  Then 
since  the  perpendicular  from  0'  on  Oq  is  equal  to  00'  sin  6', 


CARNOT'S  PRINCIPLE.  173 

it  is  easily  seen  that  the  attraction  of  B  on  the  element 
at  q  is  equivalent  to  a  force  \     P  T    ™  .  ^^  acting  at  0' 

COS  u          L/C/ 


parallel  to  qO,  together  with  a  couple  X      ~7)7y  —  •  ^an  ^ 
in  the  plane  O'qO.     In  like  manner,  the  attraction  of  B 
on  the  element  at  q'  is  equivalent  to  a  force 
Mp'r'dw      of" 
'~^s~0'    '  00'* 

acting  at  0'  parallel  to  q'O,  together  with  a  couple  in  the 
same  plane  as  the  former,  and  equal  and  opposite  to  it. 
If  the  different  parts  of  the  sphere  B'  preserve  the  same 
relative  positions  with  respect  to  one  another,  the  two 
opposite  couples  will  neutralize  one  another  (as  explained 
in  books  on  Statics),  and  need  not  be  considered.  We 
are  therefore  left  with  two  equal  forces  at  0'  which  give  a 

resultant  2\  7\T\>^~  dm  acting  at  0'  along  O'O.  Integra- 
ting, we  find  the  attraction  of  B  on  the  whole  shell 
equivalent  to  a  single  force  X  —  -7^7:,  —  —  acting  at  0'  along 

O'O.  Hence,  since  4>7ra'2r'p'  is  the  mass  of  the  shell,  the 
result  is  the  same  as  if  the  whole  shell  were  condensed 
into  a  particle  at  0'.  Treating  every  one  of  the  shells 
of  B  in  the  same  way,  we  see  that  the  attraction  between 
the  two  spheres  is  the  same  as  if  both  spheres  were  con- 
centrated into  particles  at  their  centres  ;  so  that  if  M,  M', 
be  the  masses  of  the  spheres,  and  r  the  distance  between 
their  centres,  their  mutual  attraction  will  be 

.  MM' 
*->    • 

The  formula  which  has   been  just  obtained  has  two 


174  ELEMENTARY  THERMODYNAMICS. 

very  important  applications.  In  the  first  place,  it  appears 
from  the  labours  of  astronomers,  that  the  sun  and  planets 
are  very  nearly  spherical  bodies,  each  of  which  is  composed 
of  a  number  of  spherical  shells  of  uniform  density  con- 
centric with  one  another.  In  treating  them  as  if  they 
were  concentrated  into  points  at  their  centres,  we  have 
therefore  been  making  a  near  approximation.  In  the 
second  place,  an  experiment  known  as  the  Cavendish 
experiment  has  been  performed  by  Mr  Baily,  which 
virtually  amounts  to  measuring  the  mutual  attraction  of 
two  homogeneous  spheres  of  known  masses,  from  which  it 
is  calculated  that  the  value  of  X,  in  the  c.  G.  s.  system  of 

fi'4<8 
units,  is  —  —  .     In  the  English  system  of  units,  in  which 

the  foot,  the  pound,  and  the  poundal  are  units,  X  =  -j^. 
(See  Routh's  Rigid  Dynamics  on  the  Cavendish  experi- 
ment ;  also  Everett's  Units  and  Physical  Constants.) 

It  will  be  instructive  to  calculate  the  attraction 
between  two  equal  and  similar  homogeneous  spheres  of 
a  moderate  size.  Let  the  radius  of  each  sphere  be  r 
centimetres,  and  the  density  p  grammes  per  cubic  centi- 
metre, and  let  them  be  placed  in  contact.  Then  the 
mass  of  each  sphere  being  |-7rr3/3  grammes,  and  the  dis- 
tance between  their  centres  2r  centimetres,  their  mutual 
attraction  F,  in  dynes,  will  be 


= 


=  Xf  7r2ry. 

If  the  spheres  be  of  lead,  the  value  of  p  may  be  taken  to  be 
11  J,  so  that  the  mass  of  each  sphere  =46'5  x  r3  grammes,  and 


365  x  10~7  x 


CARNOT'S  PRINCIPLE. 


175 


By  means   of  these  formulae   the  following   results   are 
calculated : 


Diameter 
of  each  sphere. 


One  metre 


One  centimetre 


One  kilometre  (1093-6 
yards) 


Mutual  attraction. 

228  dynes,  or  -23  gramme, 

or    -0005   lb.,    or    the 

2000th  of  a  lb. 
228xlO~8  dynes,  or  the 

200,000     millionth    of 

alb. 
228  xlO12   dynes,   or    230 

million     kilogrammes, 

or     5  x  108     Ibs.,     or 

229,000  tons. 


of  each  sphere. 

5800  kilogrammes, 
or  12786  Ibs.,  or 
5-7  tons. 

5 -8  grammes,  or 
•013  lb. 

5,800,000  million 
kilogrammes,  or 
5,700  million 
tons. 


25'64  centimetres,  or  lO'l  inches  One  dyne. 

1-43  metres,  or  56-5  inches  One  gramme-weight. 

2'7  metres,  or  9  feet  One  poundal. 

6'7  metres,  or  21 '9  feet  One  pound- weight. 

75.  It  appears  from  observation  that  the  heavenly 
bodies  are  very  approximately  of  the  forms  generated  by 
the  revolution  of  nearly  circular  ellipses  about  their  minor 
axes.  Again,  it  is  obvious  that  no  body,  however  hard  or 
solid,  can  be  '  perfectly  rigid/  that  is,  invariable  as  to  the 
relative  positions  of  its  parts.  For  these  two  reasons,  it  is 
not  quite  accurate  to  treat  the  heavenly  bodies  as  mere 
particles. 

To  illustrate  how  the  want  of  'perfect  rigidity,' 
combined  with  the  deviation  from  sphericity,  enables 
gravitation  to  modify  the  celestial  motions  by  means  of 
friction,  it  will  be  sufficient  to  treat  the  sun  and  planets 
as  if  each  of  them  consisted  of  two  parts,  (1)  a  rigid, 
homogeneous,  nucleus  A,  in  the  form  of  the  figure 


176  ELEMENTARY  THERMODYNAMICS. 

generated  by  the  revolution  of  a  nearly  circular  ellipse 
about  its  minor  axis,  and  (2)  an  outer  part,  or  '  tide,'  B, 
moveable  over  the  surface  of  the  nucleus  A.  Let  such  a 
body  be  held  so  that  its  centre  of  mass  is  fixed  at  G,  and 
so  that  some  particular  line,  passing  through  0  in  a 
direction  fixed  in  the  body,  is  also  fixed  in  space.  Take 
the  fixed  direction  through  0  as  axis  of  z,  and  let  a  planet 
P,  which,  in  its  effects  on  the  planet  G,  may,  for  the 
purposes  of  illustration,  be  treated  as  a  particle,  be 
prevented  from  moving  except  in  a  circular  orbit  about 
G  as  centre,  in  a  plane  perpendicular  to  Gz.  Also  suppose 
the  mechanism  by  which  these  restrictions  are  imposed  to 
be  such  that  its  attractive  and  other  forces  on  any  part  of 
G  are  insignificant,  except  near  the  axis  Gz,  and  let  these 
forces  be  so  symmetrical  that  their  moment  about  Gz  is 
constantly  zero.  Lastly,  let  there  be  no  other  bodies  near 
the  planet  G.  Then  in  G  we  practically  have  a  body 
which  is  free  to  rotate  about  a  fixed  axis  Gz  under  no 
exertal  force  but  the  attraction  of  a  single  particle  P. 

If  Gz  be  the  axis  of  figure  of  the  nucleus  A,  it  is 
obvious  that  the  attraction  exerted  on  it  by  P  will  have 
no  moment  about  Gz,  and  that  if  P  be  fixed,  the  nucleus 
will  have  an  infinite  number  of  positions  of  equilibrium, 
in  each  of  which  the  tide  B  will  be  symmetrical  about 
the  plane  through  Gz  and  P,  being  collected  into  heaps 
around  the  two  points  in  which  GP  meets  the  surface 
of  A 

If,  in  the  first  place,  when  P  and  G  are  in  this  state,  we 
move  P  forward  in  its  orbit,  its  attraction  will  carry  the 
tide  B  round  in  the  same  direction;  but  owing  to  the 
frictional  resistance  which  will  be  experienced  in  moving 
over  the  surface  of  the  nucleus  A,  the  tide  will  not 


CARNOT'S  PRINCIPLE. 


177 


generally  continue  to  be  symmetrical  with  respect  to  the 
plane  zGP.  On  this  account,  the  attraction  of  P  on  B 
will  generally  give  a  moment  about  Gz,  and  so  will  the 
frictional  action  exerted  by  B  on  A.  The  nucleus  A  will 
therefore  be  carried  round  Gz  in  the  same  direction  as  P ; 
and  it  is  clear  that  if  the  angular  velocity  of  P  in  its  orbit 
be  kept  constant,  the  nucleus  A  will  at  length  acquire  the 
same  uniform  angular  velocity  about  Gz,  and  the  tide  B 
take  up  a  fixed  position  on  A  symmetrical  with  respect  to 
the  moving  plane  zGP. 

If,  on  the  other  hand,  when  G  and  P  are  in  a  state  of 
equilibrium,  we  keep  P  fixed  and  set  the  nucleus  A  in 
motion  about  Gz,  the  tide  B  will  be  carried  partially 
round  with  it  against  the  attraction  of  P.  Hence  when 
we  abandon  G  to  the  influence  of  P,  there  will  be  a  couple 
about  Gz  by  which  the  angular  momentum  of  the  whole 
body  G  about  Gz  will  be  ultimately  reduced  to  zero,  and 
the  tide  B  brought  into  the  same  relative  position  with 
respect  to  zGP  as  before. 

Again,  if  Gz  be  at  right  angles  to  the  axis  of  figure,  it 
may  be  shown  that,  except  when  the  axis 
of  figure  passes  through  P,  the  attraction 
of  P  on  the  nucleus  A  will  exert  a  moment 
about  Gz  tending  to  make  the  equator  of 
A  pass  through  P.  When  the  axis  of  figure 
passes  through  P,  the  moment  about  Gz 
will  be  zero,  but  the  slightest  displacement 
from  this  position  would  bring  the  moment 
into  existence.  It  is  therefore  evident  that 
if  P  be  at  rest  and  the  tide  B  absent,  the 
nucleus  A  will  either  be  at  rest  in  a  state 
of  stable  equilibrium  with  its  equator 

p.  12 


178  ELEMENTARY   THERMODYNAMICS. 

passing  through  P,  or  else  will  be  rotating  about  Gz.  If 
there  be  a  tide  B,  the  rotation  will  be  checked  by  tidal 
friction,  and  A  will  finally  come  to  rest  with  its  equator 
passing  through  P. 

In  like  manner,  if  P  be  rotated  in  its  orbit  with  uniform 
angular  velocity,  it  may  be  inferred  that  A  will  be  carried 
round  with  it  and  will  ultimately  rotate  round  Gz  at  the 
same  speed  as  P,  with  its  equator  passing  through  P,  that 
is,  with  its  axis  of  figure  at  right  angles  to  GP. 

From  the  preceding  arguments,  it  is  clear  that  if  the 
fixed  axis  of  rotation  of  the  planet  G  be  at  right  angles  to 
the  axis  of  figure,  and  be  placed,  not  at  right  angles  to 
the  orbit  of  P,  but  in  it,  the  planet  G  will  ultimately  place 
its  equator  in  the  orbit  of  P,  or  its  axis  of  figure  at  right 
angles  to  that  orbit,  except  only  when  P  is  kept  at  rest 
in  either  of  the  two  points  in  which  the  axis  of  rotation 
of  G  produced  meets  its  orbit.  Hence  also  if  no  part  of 
G  be  fixed  except  the  centre  of  mass,  and  if  P  describe 
its  circular  orbit  with  any  uniform  angular  velocity  &>,  the 
planet  G  will  ultimately  set  its  axis  of  figure  at  right 
angles  to  the  orbit  of  P  and  rotate  about  that  axis  with 
uniform  angular  velocity  <w. 

It  may  therefore  be  inferred  that,  if  any  material 
system  be  protected  from  all  external  influences,  the  all- 
pervading  force  of  gravitation  will  in  time  destroy  all 
relative  mechanical  motions,  so  that  the  only  mechanical 
motions  of  the  system  will  then1  consist  of  a  constant 
velocity  of  0,  the  centre  of  mass,  in  a  straight  line,  com- 
bined with  a  constant  angular  rotation  of  the  whole 
system  about  some  straight  line  passing  through  G  in  a 
fixed  direction.  This  state,  it  will  be  easily  seen,  requires 
1  See  Art.  24. 


CARNOT'S  PRINCIPLE.  179 

that  the  centres  of  mass  of  all  the  distinct  bodies  of  which 
the  system  is  composed  should  lie  in  a  plane  at  right 
angles  to  the  fixed  straight  line  about  which  the  whole 
system  revolves.  Furthermore,  if  any  of  these  bodies  be 
bodies  of  revolution,  they  will  have  their  axes  of  figure 
parallel  to  the  axis  of  rotation. 

Now  if  0  be  the  point  where  the  axis  Oz,  about  which 
the  whole  system  rotates,  meets  the  plane  containing  the 
centres  of  mass,  and  if  G,  G',  be  the  positions  of  the  centre 


of  mass  of  any  one  of  the  distinct  bodies  which  compose 
the  system,  it  is  clear  that  the  displacement  of  the  body 
G,  due  to  the  rotation  about  Oz,  is  equivalent  to  a  transla- 
tion parallel  to  GG',  together  with  a  rotation  of  the  whole 
body  about  an  axis  through  G  parallel  to  Oz,  in  the  same 
direction  as  the  rotation  about  Oz,  and  through  an  angle 
equal  to  GOG'.  Hence  a  rotation  of  the  body  about  Oz  is 
equivalent  to  an  equal  rotation  in  the  same  direction  of 
its  centre  of  mass  G  about  Oz,  together  with  an  equal 
rotation  in  the  same  direction  of  the  whole  body  about  an 
axis  through  G  parallel  to  Oz.  The  final  state  of  the 
system  will  therefore  be  such  that : — 

12-2 


180  ELEMENTARY   THERMODYNAMICS. 

(1)  The  centres  of  mass  of  all  the  distinct  bodies  will 
lie  in  one  plane,  the  normal  to  which  is  fixed  in  direc- 
tion. 

(2)  The  axes  of  figure  of  all  those  which  are  bodies 
of  revolution  will  be  perpendicular  to  this  plane. 

(3)  In  addition  to  a  uniform  velocity  of  all  parts  of 
the  system  in  the  same  constant  direction,  the  centres 
of  mass  will  all  describe  circular  orbits  in  the  same 
direction    and    with    a    common    constant1    angular 
velocity,  w  say,  about  0,  the  centre  of  mass   of  the 
whole  system. 

(4)  Each  body  will  also  rotate  about  an  axis  through 
its  centre  of  mass  parallel  to  the  fixed  direction,  with 
the  same  constant  angular  velocity  o>,  and  in  the  same 
direction,  as  the  centres  of  mass  rotate  about  0. 

The  only  external  influence  to  which  the  solar  system 
appears  to  be  subject  is  the  radiation  of  energy  into  space. 
This  will  probably  continue  so  long  as  any  parts  are  above 
the  absolute  zero  of  temperature.  We  therefore  conclude 
that  in  the  final  state  of  the  solar  system,  every  part  will 
be  at  the  absolute  zero  of  temperature,  and  consequently, 
that  all  non-mechanical  motions  will  be  absent. 
The  final  relative  positions  of  the  bodies  forming  the  solar 
system,  and  their  final  relative  mechanical  motions,  will 
be  of  the  same  character  as  if  there  had  been  no  external 
influences. 

76.  Some  of  the  effects  of  friction  are  so  important 
that  they  merit  a  more  detailed  consideration ;  but  before 
we  can  discuss  them,  we  need  an  additional  property  of 
angular  momentum. 

1  See  Art.  24. 


CARNOT'S  PRINCIPLE.  181 

Let  0  be  any  point,  and  suppose  that  OS  is  a  straight 
line  through  0  such  that,  at  any  instant  t,  the  angular 
momentum  of  a  particle  P  about  OS  is  greater  than  about 
any  other  straight  line  passing  through  0.  Draw  any  line 
OT  through  0  making  an  angle  0  with  08,  and  let  it  be 
required  to  compare  the  angular  momenta  of  P  about  OS 
and  OZ7  at  the  time  t. 

First,  let  PQ,  the  line  of  motion  of  P  at  the  time  t  be 
parallel  to  the  plane  SOT.  Then,  it  is  evident,  PQ  will 
lie  in  a  plane  perpendicular  to  OS,  and  will  make  an 
angle  9  with  a  plane  perpendicular  to  OT.  Hence  if  p 
be  the  perpendicular  distance  of  P  from  the  plane  SOT, 
m  the  mass  of  the  particle  P,  and  v  its  velocity,  the 
angular  momentum  of  P  about  OS  will  be  mvp.  Also  if 
the  plane  through  P  perpendicular  to  OT  meet  OT  in  N, 
and  intersect  the  plane  through  PQ  parallel  to  SOT  in 


the  line  Pq,  the  perpendicular  from  N  on  Pq  will  be 
equal  to  p,  and  the  resolved  linear  momentum  of  P  in 
the  plane  NPq  will  act  along  Pq  and  be  equal  to 
mv  cos  6.  The  angular  momentum  of  P  about  OT  is 
therefore  mv  cos  6. 

Secondly,  let  the  line  of  motion  of  P  meet  the  plane 
SOT  in  R,  and  since  the  angular  momentum  of  .a  particle 


182  ELEMENTARY  THERMODYNAMICS. 

about  any  line  is  independent  of  the  position  of  the 
particle  in  its  line  of  motion,  let  us  suppose  the  particle 
to  be  at  R.  Draw  RM  and  RN  perpendiculars  on  OS 
and  OT,  and  let  the  linear  momentum  of  the  particle  at 
R  be  resolved  into  mvl  at  right  angles  to  the  plane  SOT, 
and  mvz  in  that  plane.  Then  since  the  angular  momentum 


of  a  particle  about  any  line  is  equal  to  the  sum  of  the 
moments  of  its  component  linear  momenta,  the  angular 
momentum  of  the  particle  P  about  OS  and  OT  will  be 
respectively  equal  to  mvl .  RM  and  mv^ .  RN.  But  since 
the  angles  at  M  and  N  are  right  angles,  a  circle  described 
on  OR  as  diameter  will  pass  through  R,  N,  M,  and  0. 
The  angle  NRM  is  therefore  equal  to  TOS,  or  6.  Also 
since  the  angular  momentum  about  OS  is  greater  than 
about  any  other  line  drawn  through  0,  MR  must  be  a 
diameter  of  the  circle  RNMO,  or  the  points  0  and  M 
must  coincide.  The  angle  MNR  will  therefore  be  a  right 
angle,  and  RN  =  RMcos0.  Hence  the  angular  momenta 
of  the  particle  P  about  OS  and  OT  may  be  written  in  the 
forms  mvl .  RM  and  mv-^ .  RM  cos  6. 

In  both  cases,  if  h  be  the  angular  momentum  about 


CARNOT'S  PRINCIPLE.  183 

OS,  the  angular  momentum  about  OT  will  be  h  cos  0.  It 
thus  appears  that  the  angular  momentum  of  P  about  OS 
is  a  kind  of  resultant,  from  which  the  angular  momentum 
about  any  other  line  OT  is  to  be  found  by  resolving,  exactly 
as  if  we  were  dealing  with  forces  acting  on  a  particle  at  0, 
having  a  resultant  along  OS.  Consequently,  if  (I,  m,  n)  be 
the  angular  momenta  of  any  particle  P  about  three  rect- 
angular axes  Ox,  Oy,  Oz,  and  if  OT  be  any  line  through 

0  making  angles  (a,  /3,  7)  with  these  axes,  the  angular 
momentum  of  the  particle  about  OT  will  be 

I  cos  ct  +  m  cos  ft  +  n  cos  7. 

Similarly,  if  (I',  m,  n')  be  the  angular  momenta  of  any 
other  particle  P'  about  the  same  rectangular  axes,  its 
angular  momentum  about  OT  will  be 

I'  cos  a  +  m'  cos  ft  +  n  cos  7. 

The  angular  momenta  of  the  system  of  two  particles  about 
the  axes  and  about  OT  are  therefore  respectively  equal  to 

1  +  I',  m  +  m',  n  +  ri,  and 

(I  +  V)  cos  a  +  (m  +  m')  cos  /3  +  (n  +  n')  cos  7 ; 
whence  it  easily  follows  that  if  (L,  M,  N)  be  the  angular 
momenta   of  a  finite   body  about   the  axes,  its   angular 
momentum  about  OT  will  be 

L  cos  a  +  M  cos  /9  +  .ZVcos  7. 

Thus  if  we  draw  OA,  AB,  BG,  parallel  to  the  axes,  and 
respectively  proportional  to  (L,  M,  N),  the  angular  mo- 
mentum of  the  body  about  OT  will  be  proportional  to  the 
sum  of  the  projections  of  OA,  AB,  and  BC  on  OT;  that 
is,  proportional  to  the  projection  of  OC  on  OT.  Conse- 
quently, if  we  denote  V.L2  +  M2  +  N2  by  H,  and  the  angle 
between  OC  and  OT  by  ty,  the  angular  momentum  about 


184 


ELEMENTARY  THERMODYNAMICS. 


OT  will  be  H  cos  ^.     We  see  therefore  that  the  angular 
momentum  about  OC  is  to  be  regarded  as  a  resultant,  and 


the  angular  momentum  about  any  other  line  OT  found  by 
resolving,  exactly  as  if  we  were  dealing  with  forces  acting 
on  a  particle  at  0  and  having  a  resultant  along  OC. 

It  should  be  noticed  that  the  straight  lines  OS,  OS', 
about  which  the  angular  momenta  of  two  particles  P,  P', 
are  maxima,  will  not  generally  coincide  with  one  another, 
or  with  OC. 

77.  The  following  important  propositions  are  collected 
from  the  first  chapter,  and  will  be  required  presently. 

(1)  The  angular  momentum  of  a  body  (or  system  of 
bodies),  about  any  fixed  line  Oz,  is  equal  to  the  angular 
momentum  about  a  moveable  parallel  line  through  G,  the 
centre  of  mass,  together  with  what  the  angular  momentum 
about  Oz  would  be  if  the  whole  body  (or  system)  were 
condensed  into  a  single  particle  at  G. 
These  two  parts  are  usually  referred  to  as  the  angular 
momenta  of  rotation  and  translation,  respectively. 


CARNOT'S  PRINCIPLE.  185 

(2)  The  angular  momentum  of  rotation  depends  only 
on  the  velocities  relative  to  the  centre  of  mass. 

(3)  If  we  assume  that  the  internal  forces  of  the  body 
(or  system)  consist  of  a  set  of  equal  and  opposite  reactions, 
the  rate  at  which  the  angular  momentum  of  the  body  (or 
system)  about  any  fixed  line  increases  with  the  time,  will 
be  equal  to  the  moment  about  that  line  of  the  external 
forces. 

(4)  If  the  internal  forces  consist  of  a  set  of  equal  and 
opposite  reactions,  the  rate  at  which  the  angular  momen- 
tum about  a  moveable  line  GN,  through  G  parallel  to  any 
fixed  line  Oz,  increases  with  the  time,  will  be  equal  to  the 
moment  about  GN  of  the  external  forces  ;  and  the  rate  at 
which  the  angular   momentum  of  translation  about    Oz 
increases  with  the  time  will   be  equal   to  the   moment 
about  Oz  of  a  system  of  forces  applied  at  G  equal  and 
parallel  to  the  external  forces.     Hence  when  there  are  no 
external  forces,  the  angular  momenta,  both  of  translation 
and  rotation,  will  remain  constant  with   respect   to   any 
fixed  line. 

(5)  If  w  be  the  angular  velocity  of  a  body  in  which 
there  are  no  mechanical  vibrations,  and  G  the  moment  of 
inertia  about  the  axis  of  rotation,  the  angular  momentum 
about  this  axis  is  Cw. 

(6)  If  M  be  the  mass  of  the  body,  and  V  the  velocity 
of  its  centre  of  mass,  its  mechanical  kinetic  energy  will  be 


78.  To  illustrate  the  effects  of  friction  and  mechanical 
vibrations  in  modifying  the  mechanical  motions  of  a  body 
which  is  subject  to  no  external  influences,  let  us  first  take 
the  case  of  an  unelectrified  body  of  the  form  generated  by 


186  ELEMENTARY   THERMODYNAMICS. 

the  revolution  of  an  ellipse  about  its  minor  axis,  and  either 
homogeneous  throughout,  or  symmetrical  with  respect  to 
every  plane  through  the  axis  of  revolution  and  such  that 
the  moment  of  inertia  about  that  axis  is  greater  than 
about  any  perpendicular  axis.  Also  let  the  body  be 
extremely  hard  and  unyielding,  and  the  frictional  effects 
and  mechanical  vibrations  consequently  so  minute,  that, 
for  the  purpose  of  calculating  the  angular  momentum  and 
mechanical  kinetic  energy  at  any  instant,  we  may,  without 
sensible  error,  suppose  the  relative  positions  of  all  the 
parts  to  be  invariable,  and  represent  the  mechanical 
motions  of  the  body  by  the  motion  of  the  centre  of  mass 
combined  with  a  rotation  about  an  axis  passing  through 
the  centre  of  mass. 

We  assume,  as  usual,  that  the  internal  forces  consist 
of  a  set  of  equal  and  opposite  reactions,  in  consequence  of 
which  the  body  will  possess  some  important  properties  : — 

(1)  The  velocity  V,  of  the  centre  of  mass,  will  remain 
constant. 

(2)  The  angular  momentum  of  rotation  with  respect 
to  any  fixed  straight  line  will  be  constant ;  in  other  words, 
as  the  body  moves  about,  the  angular  momentum  about  a 
straight  line  drawn  through  the  centre  of  mass  in  a  fixed 
direction,  will  be  invariable. 

(3)  Hence,  as  the  body  moves  about,  the  resultant 
angular   momentum   at    G,  the   centre    of  mass,  will  be 
constant,  both  in  magnitude  and  direction. 

Again,  if  in  any  state  P,  when  the  body  is  subject 
to  no  external  influences,  we  apply  external  forces  so 
as  to  bring  it  to  a  state  of  mechanical  rest,  it  will  be 
seen,  on  referring  to  Art.  25,  that,  on  account  of  the 
hardness  of  the  body,  the  mechanical  work  thus  obtained 


CARNOT'S  PRINCIPLE.  187 

from  it  may  be  practically  equal  to  the  mechanical 
kinetic  energy  in  the  state  P.  Suppose  then  that  we 
make  the  body  undergo  the  following  complete  cycle  of 
operations : — 

(1)  Let   it   pass,   under   the   action   of  no   external 
influences,  from  any  state  P,  in  which  the  temperature 
is  uniform,  to  any  other  state  Q.     In  this  operation,  the 
entropy  will  increase. 

(2)  Let   the   body  be   then  reduced   to  a  state   of 
mechanical  rest  in  such  a  way  that  the  mechanical  work 
obtained  is  practically  equal  to  Tq,  the  mechanical  kinetic 
energy  in  the  state  Q.     Also  since  the  temperature  in  the 
state  Q  will  not  be  uniform,  let  sufficient  time  be  allowed 
for  it  to  become  so  before  proceeding  to  the  next  opera- 
tion. 

On  account  of  the  irreversible  equalization  of  tempera- 
ture which  has  just  taken  place,  the  entropy  will  have 
again  increased. 

(3)  Next,  let  the  temperature  be  raised  or  lowered 
until  the  temperature  of  the  initial  state  P  is  attained, 
the   angular  momentum  and  mechanical  kinetic  energy 
both  being  kept  zero. 

(4)  Lastly,  let  the  body  be  brought  to  its  initial  state 
by  doing  an  amount  of  work  upon  it  equal  to  Tp,  the 
mechanical  kinetic  energy  in  the  state  P.     This  will  not 
alter  the  entropy  of  the  body. 

Since  the  entropy  has  the  same  value  at  the  end  of  a 
complete  cycle  as  at  the  beginning,  it  follows  that  in  the 
third  operation  the  entropy  must  have  decreased.  In  this 

operation,  therefore,  we  have  /-^  negative.  Hence  since 
it  is  a  priori  evident  that  the  operation  may  be  performed 


188 


ELEMENTARY  THERMODYNAMICS. 


in  one  or  other  of  two  ways,  either  so  as  to  make  every 
element  of  heat,  dQ,  positive,  or  so  as  to  make  every 
element  negative,  we  conclude  that  the  total  amount  of 
heat  absorbed  will  be  negative.  Denoting  it  by  —  Q, 
where  Q  is  a  positive  quantity,  the  principle  of  energy 
gives 

Tp-Tg-Q  =  0, 

Tq-Tf  =  -Q, 

from  which  we  conclude  that  the  mechanical  kinetic 
energy  continually  decreases  until  the  final  invariable 
state  is  reached. 

Let  us  consider  the  motion  at  any  instant  t,  when  G  is 
the  position  of  the  centre  of  mass,  Gz  of  the  axis  of  figure, 
and  Gz'  of  the  axis  about  which  the  body  is  then  rotating. 
Take  Gx  at  right  angles  both  to  Gz  and  Gz' ;  and  take 


Gy,  Gy'  in  the  plane  zGz,  perpendicular  respectively  to 
Gz  and  Gz'.  Then,  if  o>  be  the  angular  velocity  of  the 
body  at  the  time  t,  and  C"  the  moment  of  inertia  about 
Gz,  the  angular  momentum  about  Gz  will  then  be  <7o> ; 
about  Gy',  —  o>  ^mz'y' ;  and  about  Gx,  —  to  ^.mz'x,  which 


CARNOT'S  PRINCIPLE.  189 

is  clearly  zero.  Thus  the  axis,  GK,  of  resultant  momen- 
tum at  G,  lies  in  the  plane  zGz',  and  the  resultant  momen- 
tum H  is  such  that 


Also  the  mechanical  kinetic  energy,  T,  at  the  time  t,  is 
given  by 


where  M  is  the  mass  of  the  body  and  F  the  velocity  of  its 
centre  of  mass. 

Hence,  since  H  and  F  are  constant  while  T  continually 
diminishes  until  the  ultimate  state  is  attained,  we  see  that 


must  continually  increase  up  to  the  final  state. 


On  account  of  the  symmetrical  form  of  the  body,  this 
result  can  be  exhibited  in  a  simpler  form. 
If  we  denote   the   angle   between   Oz  and    Oz,  or  the 


190  ELEMENTARY  THERMODYNAMICS. 

equal  angle  between  Oy  and  Oy',  by  6,  the  coordinates  of 
any  point  P  will  be  connected  by  the  relations 

y'  =  y  cos  6  —  z  sin  6  ] 
z  =  y  sin  6  +  z  cos  6 }  ' 
Thus,  since  2mzy  is  clearly  zero, 

=  2m  O2  +  y~}  +  sin2  6 .  2m  (z2  -  y2) 

=  2m  (x>  +  t/2)  +  sin2  6  .  2m  {(z2  +  x2)  -  (yz  +  #2)} 


where  C  is  the  moment  of  inertia  about  the  axis  of  figure, 
and  A  that  about  any  perpendicular  axis. 

Again,  2mz'y'  =  sin  6  cos  6  .  2m  (y*  —  z1) 


Hence    C'2  +  (2mz'yJ  =  &  cos2  0  +  A2  sin2  0 
=  (A+C)C'-AC. 

Thus  we  see  that,  until  the  ultimate  state  is  reached,  the 

AC 
value  of  A  +  C  —  ^-  must  continually  increase.     Thence 

L/ 

it  follows  that  C'  continually  increases,  and  therefore, 
since  w2  {(A  +  (7)  C'  —  AC}  remains  constant,  that  a>  con- 
tinually decreases. 

Now  since  C'  =  C  +  (A  —  C)  sin2  6,  where  A  is  evidently 
less  than  C,  the  moment  of  inertia  about  the  axis  of  figure 
will  be  greater  than  about  any  other  axis  through  G,  and 
the  moment  of  inertia  about  any  axis  through  G  will 

decrease  as  6  increases  up  to  J  The  effect  of  friction  will 
therefore  be  to  bring  the  axis  of  rotation  nearer  to  the 


CARNOT'S  PRINCIPLE.  191 

axis  of  figure ;  and  when  they  once  coincide,  they  will 
not  again  separate,  so  long  as  external  influences  are 
absent. 

Furthermore,  the  instantaneous  axis  of  rotation,  Gz', 
can  only  coincide  with  the  axis  of  resultant  angular 
momentum  at  G  when  the  angular  momentum  about  any 
straight  line  through  G  at  right  angles  to  Gz'  is  zero. 

This  can  only  take  place  when  6  =  0  or  0  =  -^,  so  that  the 

axis  of  rotation  must  then  either  coincide  with  the  axis  of 
figure  or  lie  in  the  equator. 

Hence,  since  friction  will  prevent  the  axis  of  rotation 
from  getting  into  the  equator,  we  conclude  that  the  effects 
of  friction  will  not  cease  until  the  axis  of  rotation  and  the 
axis  of  figure  coincide  with  the  axis  of  resultant  angular 
momentum  at  G. 

If  we  were  to  observe  the  motion  for  a  short  time  only, 
the  effects  of  friction  would  be  imperceptible,  and  C"  and 
6  would  be  sensibly  constant,  so  that  the  axis  of  rotation 
would  appear  to  describe  a  cone  in  the  body  about  the 
axis  of  figure.  Consequently,  if  an  ideal  sphere  be  supposed 
carried  about  fixed  in  the  body  with  its  centre  at  G,  the 
axis  of  rotation  would  intersect  its  surface  in  a  circle.  On 
watching  the  motion  long  enough,  it  would,  however,  be 
seen  that  the  curve  was  not  a  perfect  circle,  but  a  nearly 
circular  spiral  gradually  approaching  the  point  in  which 
the  axis  of  figure  meets  the  sphere. 

To  find  the  motion  as  it  would  appear  for  a  short 
time,  we  must  first  find  the  position  of  GK  in  the  plane 
zGz'. 

Since  when  the  angle  zGz'  is  6,  the  angular  momentum 
about  Gz!  is  C'w,  and  about  Gy',  —  CD  (C  —  A)  sin  6  cos  0,  the 


192 


ELEMENTARY  THERMODYNAMICS. 


axis  GK  will  lie  on  the  side  of  Gz'  remote  from  Gy',  and 
make  an  angle  ty'  with  Gz',  such  that 

,  _  (C -  A)  sin  6  cos  6 
C 


C  cos2  0  +  A  sin2  0 
(G-A)sm0cos0 
(C-A)cos20  +  A 
<  tan  0. 

K 


Thus  GK  lies  between  Gz'  and  Gz. 

Again,  if  ty  be  the  angle  between  GK  and  Gz,  we  have 


and 


tan  0  —  tan  -ft' 
l+tan0tan>jr 


C 


CARNOT'S  PRINCIPLE.  193 

Hence  also 

A  sin  6 
sm-vjr  = 


VC2  cos2  d  +  A-  sin2  0  ' 
a>(C-A)sm0cos0 


-A  )sm0cos0 


Since  we  only  require  the  motion  relative  to  G,  let  us 
take  G  to  be  at  rest  ;  and  suppose  a  fixed  ideal  sphere  of 
unit  radius  described  with  its  centre  at  G.  Let  GK,  the 
axis  of  resultant  angular  momentum  at  G,  meet  the  sphere 
in  the  fixed  point  K,  and  let  Gz,  Gz',  the  positions  at  the 
time  t  of  the  axes  of  figure  and  rotation,  meet  it  in  the 
points  z,  z  .  Then  if,  at  the  consecutive  instant  t  +  T,  the 
axes  of  figure  and  rotation  meet  the  sphere  in  the  points 
zly  z\,  these  points  will  lie  in  a  plane  through  GK,  the 
arcs  ZZ-L,  z'z\,  will  be  at  right  angles  to  the  arc  zKz',  and 


zzt  will  be  equal  to  car  sin  0.     The  angle  zKz  will  there- 

,       «OT  sin  0         ,  L ,  ,  ,  L    <OT  sin  0  . 

fore  be  equal  to      .    -  -;  and  the  arc  z  z  x  to  — .-  -  -    sin  A|T  , 
sin  ty  sin  ijr 

C1  —  A 

or  WT  —    — sin  0  cos  0.     Hence  the  plane  zGz'K  rotates  in 
A. 

space  about  GK,  in  the  same  direction  as  to,  with  angular 

velocity   °>sin,S,  or  .  VC" <™°  9  +  4!g°g.     Also,  since 
sm^/r  ^ 

the  same  point  of  the  body  is  situated  at  /  at  the  times  t, 

t  +  T,  the  plane  zGz'K  rotates  in  the  body  about  the  axis 

P.  13 


194  ELEMENTARY   THERMODYNAMICS. 

of  figure,  in  the  same  direction  as  w,  with  angular  velocity 

C-A 

w cos  6. 

A. 

When    6  is   indefinitely  small,   these   angular   velocities 
become,  respectively,  -j-  and  — —  to,  so  that  their  ratios 

to  <w  are  both  finite. 

If,  secondly,  we  have  a  body  so  fluid  that  it  can  assume 
a  form  symmetrical  with  respect  to  every  plane  containing 
any  given  axis  passing  through  its  centre  of  mass,  and  if, 
after  setting  it  in  rotation  in  any  way,  we  protect  it  from 
all  external  influences ;  it  is  evident  that  the  parts  of  the 
body  through  which  the  axis  of  rotation  will  ultimately 
pass,  will  not  be  the  same  whatever  may  be  the  state 
of  the  body  when  first  abandoned  to  itself.  For,  if  the 
body  be  unelectrified  and  be  held  until  it  assumes  a  figure 
of  revolution  about  the  axis  of  rotation,  it  is  clear  that,  in 
the  invariable  state  assumed  after  the  external  influences 
have  been  removed,  the  axis  of  rotation  will  pass  through 
the  same  parts  of  the  body  as  when  the  body  is  first  left  to 
itself.  Consequently,  if  we  take  any  two  such  cases,  in 
which  the  body  is  rotating  about  different  axes  when  first 
abandoned  to  itself,  the  axis  of  rotation  cannot  pass  through 
the  same  parts  of  the  body  in  the  two  final  states. 
Again,  if  the  body  be  allowed  to  assume  an  invariable 
state,  and  then  be  acted  on  for  a  time  by  external  causes, 
the  axis  of  rotation  in  the  ultimate  invariable  state  will 
not  generally  pass  through  the  same  parts  of  the  body  as 
before  the  disturbing  forces  came  into  operation.  Hence, 
since  the  body  will  be  symmetrical  with  respect  to  every 
plane  through  the  axis  of  rotation  in  every  invariable  state 


CARNOT'S  PRINCIPLE.  195 

assumed  under  the  action  of  no  external  influences,  the 
disturbing  forces  will  generally  have  the  effect  of  altering 
the  relative  positions  of  the  parts  of  the  body. 

79.  The  surface  of  the  earth  is  about  three-fourths 
covered  with  a  comparatively  shallow,  fluid  ocean :  the  rest 
of  the  mass  may  be  treated  as  'rigid,'  that  is,  as  if  the 
relative  positions  of  its  parts  were  invariable.  The  rigid 
nucleus  appears  to  be  practically  symmetrical  with  respect 
to  every  plane  through  an  axis  Gz,  where  G  is  the  centre 
of  mass ;  while  the  ocean  can  change  its  place  on  the 
surface  to  suit  the  forces  which  act  upon  it.  We  therefore 
conclude  that  if  the  earth  be  protected  from  all  external 
influences,  or  be  merely  allowed  to  radiate  energy  into 
space,  it  will  be  intermediate  in  behaviour  between  the 
two  bodies  considered  in  the  last  article.  Thus,  on  account 
of  the  preponderating  mass  of  the  rigid  nucleus,  the 
ultimate  axis  of  rotation  will  always  be  very  nearly 
coincident  with  Gz,  the  axis  of  figure ;  and  on  account 
of  the  fluidity  of  the  ocean,  the  rate  at  which  the  axis 
of  rotation  approaches  Gz  will  be  comparatively  rapid. 
In  reality,  the  earth  is  acted  on  by  the  attractions  of 
the  sun,  moon,  and  planets ;  and  as  these  bodies  are  not 
situated  in  the  plane  of  the  earth's  equator,  their  attractions 
will  exert  a  small  moment  about  an  axis  through  G  per- 
pendicular to  Gz.  Thus  if  the  whole  of  the  earth  were  as 
rigid  as  the  nucleus,  the  axis  of  rotation  might,  in  time, 
separate  considerably  from  the  axis  of  figure ;  but  the 
fluidity  of  the  ocean  will  have  a  contrary  tendency  to 
keep  the  deviation  of  these  axes  small,  or,  as  we  may 
express  it,  as  the  axis  of  rotation  moves  about  in  space,  it 
will  be  followed  by  the  axis  of  figure. 

13—2 


19()  ELEMENTARY  THERMODYNAMICS. 

The  position  of  the  sea  with  respect  to  the  land  is 
determined  to  a  small  extent  by  the  inequalities  of  the 
surface  of  the  nucleus,  but  chiefly  by  the  centrifugal  force 
due  to  the  diurnal  rotation,  combined  with  the  attraction 
of  the  rigid  nucleus  and  with  the  attraction  of  the  water 
on  itself.  The  principal  variable  elements  on  which  the 
form  of  the  sea  depends,  are  therefore  the  position  of  the 
axis  of  rotation  with  respect  to  the  rigid  nucleus,  and  the 
amount  of  the  angular  rotation.  If  the  axis  of  rotation 
were  inclined  at  any  considerable  angle  to  Gz,  the  axis  of 
figure  of  the  nucleus,  these  invariable  elements  would  be 
undergoing  continual  changes,  and  in  consequence,  the 
ocean  would  be  perpetually  altering  its  distribution  011 
the  surface  of  the  earth,  in  some  places  flooding  the  land, 
in  others,  leaving  large  tracts  of  sea  dry.  That  such  an 
event  does  not  occur,  evidently  proves  the  ocean  capable 
of  keeping  the  two  axes  near  together  in  spite  of  the 
efforts  of  the  sun,  moon,  and  planets,  to  make  them 
separate. 

At  present,  when  the  axis  of  rotation  is  practically 
coincident  with  Gz,  the  axis  of  figure,  and  the  angular 

velocity  equal  to  ~^~Q^  >  tne  surface  of  the  earth  is  very 

approximately  that  generated  by  the  revolution  of  an 
ellipse  about  its  minor  axis,  the  centre  of  the  ellipse 
being  placed  at  G,  and  the  minor  axis  along  Gz.  The 
equation  to  this  surface,  in  polar  coordinates,  is 

r  =  a(l-ecos20), 

where  G  is  the  origin,  Gz  the  initial  line,  and  e  a  constant 
which  may  be  taken  to  be  ^.1 

1  Everett's  'Units,'  or  Herschell's  'Astronomy.' 


CARNOT'S  PRINCIPLE. 


197 


If  the  axis  of  rotation  were  not  identical  with  the  axis 
of  figure,  the  form  of  the  surface  would  be  different.  It 
is  not  within  the  province  of  thermodynamics  to  attempt 
an  exact  determination  of  the  form  that  would  be  then 
assumed ;  but,  merely  to  obtain  some  rough  general  ideas, 
we  will  suppose  that  when  the  angle  between  Gz',  the  axis 
of  rotation,  and  Gz,  the  axis  of  figure,  is  small,  and  the 
angular  velocity  the  same  as  before,  the  equation  to  the 
surface  is  still  r  =  a  (1  -  e  cos2  6),  but  with  Gz  for  the 
initial  line. 

Take  a  plane  section  through  Gz  and  Gz1,  and  let  any 
line  through  G  in  this  plane,  making  an  angle  6  with  Gz 
on  the  other  side  of  Gz',  meet  the  original  surface  in  P 
and  the  new  surface  in  Q.  Then  if  a  be  the  small  angle 
zGz, 

GP  =  a(l-e  cos2  0), 

GQ  =  a  [1  -  e  cos2  (6  +  a)]. 


Hence  the  height  the  sea  will  rise  at  P  on  account 
of  a  small  displacement  of  the  axis  of  rotation  from 
Gz,  will  be 

PQ  =  ae  {cos2  0  -  cos2  (0  +  a)} 
=  aea  sin  20. 


198  ELEMENTARY   THERMODYNAMICS. 

The  distance  zz'  being  aa,  we  obtain 

PQ  =  sin  20 

zz  ~   295    ' 
Thus,  for  a  given  value  of  zz' ,  PQ  will  be  a  maximum 

when  6  =  -.  ,  its  value  being  then 

zz1 
295' 

If  zz  be  a  kilometre,  the  maximum  value  of  PQ  will  be 
about  34  metres ;  if  zz'  be  a  mile,  the  maximum  value  of 
PQ  will  be  about  18  feet. 

From  this  calculation,  it  appears  that  a  slight  dis- 
placement of  the  axis  of  rotation  from  the  axis  of  figure 
would  cause  a  considerable  change  in  the  mean  level  of 
the  sea  in  places  in  latitude  40°  to  45°.  This  effect  might 
be  looked  for  in  low-lying  districts  like  the  eastern  parts 
of  England  or  the  country  round  the  sea  of  Azof.  No 
measurable  oscillation  in  the  mean  level  of  the  sea  being 
known  to  exist,  we  are  forced  to  conclude  that  the  distance 
between  the  extremities  of  the  axes  of  figure  and  rotation 
must  always  be  very  small. 

80.  We  will  next  consider  the  influence  of  tidal 
friction  in  modifying  the  orbits  of  the  heavenly  bodies 
and  the  angular  velocities  with  which  they  rotate  about 
their  axes.  This  subject  appears  to  have  been  first 
noticed  by  Kant  and  Prof.  J.  Thomson1,  and  more  ex- 
plicitly still,  by  Mayer.  A  few  years  after  this  (1854),  it 
was  pointed  out  by  Helmholtz  that  the  fact  that  the 
moon  always  turns  the  same  face  to  the  earth,  is  to  be 
accounted  for  by  the  tides  produced  by  the  earth  in  the 
1  Tail's  '  Sketch  of  Thermodynamics.' 


CARNOT'S  PRINCIPLE.  199 

moon  while  it  was  in  a  more  fluid  state  than  at  present. 
Of  late  years,  the  effects  of  tidal  friction  have  been 
studied  by  Prof.  G.  H.  Darwin1. 

To  make  the  discussion  simple,  let  us  imagine  two 
separate  bodies  S,  E,  with  no  other  body  near  them  ;  and, 
to  shorten  the  verbiage,  let  us  suppose  that  their  common 
centre  of  mass  is  at  rest.  Let  S,  E,  be  the  centres  of 
mass  of  the  two  bodies  at  any  instant  t ;  S',  E',  at  a 
consecutive  instant  t  +  r.  Then  since  SE,  S'E',  pass 
through  the  common  fixed  centre  of  mass  G,  the  four 
points  (S,  S',  E,  E'}  lie  in  one  plane.  Suppose,  next,  that 
the  masses  and  mechanical  motions  of  the  two  bodies  are 
symmetrical  with  respect  to  this  plane  at  the  time  t,  and 
let  the  plane  be  taken  to  be  that  of  the  paper.  Then 
clearly  the  centres  of  mass  will  continue  to  lie  in  the 
plane  of  the  paper,  and  the  masses  and  mechanical 
motions  to  be  symmetrical  with  respect  to  it,  so  that  we 
may  represent  the  mechanical  motions  of  each  body  at 
any  instant  with  respect  to  its  centre  of  mass  by  a 
rotation  about  an  axis  through  that  centre  of  mass  at 
right  angles  to  the  plane  of  the  paper,  combined  with 
certain  irregular  mechanical  motions  belonging  to  tides. 
Again,  let  the  tides  be  at  all  times  so  small  and  the  states 
of  the  bodies  so  stable  that  the  axes  of  rotation  may  be 
considered  fixed  in  the  bodies  and  the  moment  of  inertia 
of  each  body  about  its  axis  constant.  Suppose  also  that, 
except  for  the  tides,  each  body  would  be  symmetrical  with 
respect  to  every  plane  through  its  axis.  Lastly,  let  the 
strata  of  equal  density  in  both  bodies  be  nearly  spherical 
shells  concentric  with  the  centre  of  mass. 

If,  at  any  instant  t,  the  tides  of  the  body  E  be  heaped 
1  Thomson  and  Tait's  '  Natural  Philosophy,'  Vol.  n. 


200  ELEMENTARY   THERMODYNAMICS. 

up  symmetrically  about  the  two  points  in  which  the  line 
SG E  meets  the  surface  of  E,  the  attraction  of  S  on  E  will 
then  have  no  moment  about  the  axis  through  E.  Hence 


if  we  assume  that  the  only  forces  which  modify  the 
angular  momentum  of  the  body  E  are  due  to  the  attrac- 
tion of  8,  it  will  follow  that,  at  the  time  t,  the  rate  at 
which  the  angular  momentum  of  E  about  its  axis  is 
increasing  with  the  time,  is  zero.  Consequently,  if  at  the 
time  t,  E  is  moving  as  a  rigid  body  with  angular  velocity 

of  rotation  a)E ,  we  shall  have  -j,E-  =  0.     Thus  so  long  as  E 

moves  like  a  rigid  body  with  its  tides  symmetrical  about 
the  line  SGE,  its  angular  velocity  a>E  about  its  axis  will 
remain  constant ;  and  therefore  the  orbital  angular  velocity 
of  the  point  E  about  G,  or,  what  is  the  same  thing,  of  G 
about  E,  will  be  constant  and  equal  to  WK.  The  orbit  of 
E  about  G  will  then  evidently  be  a  circle  whose  radius 
depends  on  WE. 

If  the  orbit  of  E  about  G  be  not  circular,  the  orbital 
angular  velocity  of  E  will  not  be  constant :  if  the  orbit 
be  circular  but  not  of  the  proper  dimensions,  the  orbital 
angular  velocity  will  be  constant  but  not  equal  to  WK.  To 
illustrate  what  would  happen  in  either  of  these  cases,  let 
us  suppose  that  at  the  time  t,  the  orbital  angular  velocity 
of  E  about  G  is  less  than  WF,  or  equal  to  a>E  and  decreasing. 
Then  in  a  short  time,  the  tides  will  have  been  carried  in 
front  of  the  line  SGE;  while,  on  the  other  hand,  the 


CARNOT'S  PRINCIPLE.  201 

attraction  of  S  will  tend  to  bring  them  back  to  that  line. 
The  attraction  of  S  will  therefore  prevent  them  from  fully 


participating  in  the  rotation  of  E  about  its  axis,  so  that 
they  will  act  as  a  friction  brake  on  the  motion.  Similarly, 
if  the  orbital  angular  velocity  at  the  time  t  be  greater 
than  a>£,  or  equal  to  a)£  and  increasing,  the  tides  will  soon 
be  left  behind  the  line  SE,  while  the  attraction  of  S  will 
tend  to  drag  them  on.  Thus  they  will  again  have  a 
frictional  action,  but  this  time  they  will  accelerate  the 
rotation  of  E  on  its  axis. 

It  will  thus  be  seen  that  the  ultimate  effect  of  the  tides 
on  the  system  will  be  to  make  it  move  as  rigid,  whether 
the  two  bodies  are  ultimately  fused  into  one,  or  are  merely 
in  contact,  or  whether  they  continue  to  move  as  distinct 
bodies  at  a  distance  from  one  another  under  the  influence 
of  gravitation  alone.  In  any  case,  all  parts  of  the  system 
will  ultimately  describe  circular  orbits  about  G  with  the 
same  constant  angular  velocity. 

Suppose  now  that,  at  the  time  t,  the  two  bodies  are 
rotating  about  their  axes  without  mechanical  vibrations, 


with  angular  velocities  &>.«,,  o>F,  respectively;  and  the  points 
8,  E,  about  G  with  common  angular  velocity  o>.  Also  let 
the  masses  of  the  two  bodies  be  (8,  E) ;  their  moments  of 


202  ELEMENTARY   THERMODYNAMICS. 

inertia  about  their  axes  through  S  and  E,  (Cs>  C£)  ;  and 
denote   the   distances  G8,  GE,  by  R   and  r.     Then  the 
angular  momentum   H,  of  the   system   about   the   axis 
through  G,  perpendicular  to  the  paper,  is 
H  =  S&  +  Ei* 


Again,  if  we  assume  that  the  only  parts  of  the  energy 
of  the  system  that  can  vary,  are  the  mutual  potential 
energy  of  the  two  bodies,  and  the  mechanical  and  non- 
mechanical  kinetic  energies,  it  can  easily  be  proved,  by 
Carnot's  principle,  that  so  long  as  frictional  actions  exist, 
the  sum  of  the  mutual  potential  energy  of  the  two  bodies, 
and  of  the  mechanical  kinetic  energies,  must  continually 
diminish.  In  popular  language,  this  would  be  expressed 
by  saying  that  the  mechanical  kinetic  energies  and  mutual 
potential  energy  are  partially  converted,  by  tidal  friction, 
into  the  non-mechanical  kinetic  energy  of  '  heat.' 

To  facilitate  calculation,  let  us  suppose  that  at  the 
time  t,  the  points  S,  E  are  describing  circular  orbits  about 
the  point  G.  Then  since  the  mutual  attraction  between 
$  and  E  is  approximately  the  same  as  if  they  were  con- 
centrated into  particles  at  their  respective  centres  of  mass, 
we  have,  nearly, 


<«>• 


But  since  SR  =  Er  =  x  (say), 

we  find 


and 


CARNOT'S  PRINCIPLE.  203 


and  (S&  +  Er>)  o,  = 


Again,  the   sum   of  the  mechanical  kinetic  energies   of 
translation  at  the  time  t  is 


(50); 
and  lastly,  the  mutual  potential  energy  of  the  two  bodies  is 


where  J?0  and  r0  are  the  values  of  ,R  and  r  in  the  standard 
state. 

If  the  two  bodies  preserve  their  properties,  and  are  not 
in  contact  in  the  final  state,  so  that  they  are  compelled  to 
describe  their  orbits  about  G  by  the  force  of  gravity  alone, 
it  is  obvious  that  the  formulae  just  obtained  will  apply  to 
the  final  state  as  well  as  at  the  time  t. 

Now  let  us  make  the  supposition  that  at  the  time  t, 
the  points  E  and  S  are  rotating  about  G  in  the  positive 
direction,  or  that  &>  is  positive.  This  will  not  detract  from 
the  generality  of  our  conclusions.  Furthermore,  let  us 
denote  the  final  value  of  &>  by  H,  and  take  the  three  follow- 
ing illustrative  cases  of  the  ultimate  effects  of  the  tides. 
(1)  Let  H,  and  consequently  fl,  be  positive,  and  sup- 
pose both  ws  and  w£  greater  than  fl.  Then  tidal  friction 
will  diminish  both  (as  and  <aE,  so  that  (SB?  +  Ei*}  o>  will 
increase.  Hence,  by  equation  (49),  R  +  r  will  increase, 
and  therefore,  by  equation  (48),  <a  will  decrease,  or  the 
periodic  time  be  lengthened.  Again,  we  see  that  the 
mechanical  kinetic  energies  of  translation  and  rotation 


204  ELEMENTARY   THERMODYNAMICS. 

decrease,  the  diminution  being  accounted  for,  partly  by 
the  frictional  generation  of  non-mechanical  kinetic  energy, 
and  partly  by  the  increase  of  the  mutual  potential  energy 
of  the  two  bodies. 

(2)  Let  H  and  O  both  be  positive,  and  both  cos  and 
a)E  less  than  fi.     In  this  case,  tidal  friction  will  increase 
both   <BA,  and    a)f:;    and   consequently  diminish   R  +  r,  or 
draw  the  bodies  nearer  together  and  shorten  the  periodic 
time.     The  mechanical  kinetic  energies  of  translation  and 
rotation    will    both    increase ;    and    the    non- mechanical 
kinetic   energy  generated  by  tidal   friction  will  be  pro- 
duced entirely  at  the  expense  of  the  mutual  potential 
energy  of  the  two  bodies. 

(3)  If  H  be  negative,  fl  will  also  be  negative,  or  the 
two  centres  of  mass  will  ultimately  describe  their  orbits 
about  G  in  the  negative  direction.     This  result  would  be 
brought  about  in  the  following  manner.     The  two  bodies 
would   be    drawn   together  by   tidal   friction    until    they 
actually   came   into   contact.      If  the   directions   of    the 
orbital  motions  were  not  reversed  when  the  bodies  sepa- 
rated, they  would  again  be  drawn  together ;  and  so  on. 

It  might,  however,  happen  that  the  two  bodies  are  partially 
or  wholly  fused,  or  even  converted  into  vapour,  by  the 
violence  of  the  grinding  or  impulsive  actions  which  take 
place  when  they  come  together.  But  whether  they  are 
fused  into  one,  or  preserve  their  identities,  or  are  broken 
up  into  several  parts,  the  whole  system  will  ultimately 
rotate  about  the  axis  through  G  like  one  rigid  body. 

81.  In  the  solar  system,  almost  the  whole  of  the 
angular  momentum  is,  at  present,  in  one  direction.  Hence 
if  we  neglect  the  attraction  of  the  fixed  stars,  it  follows 


CARNOT'S  PRINCIPLE.  205 

that,  when  the  motions  are  ultimately  reduced  to  regularity, 
the  whole  system  will  rotate  together  in  this  direction. 

To  obtain  a  rough  estimate  of  the  mutual  actions  of 
the  sun  and  earth,  let  us  consider  the  two  bodies  S,  Ey  of 
Art.  80,  calling  S  the  sun  and  E  the  earth  ;  and,  for  the 
sake  of  simplicity,  let  us  imagine  all  other  bodies  to  be 
absent.  Then  tidal  friction  will  tend  to  make  the  orbits 
of  S  and  E  perfectly  circular,  and  also  to  cause  the  two 
bodies  to  turn  the  same  faces  to  one  another.  These 
effects  will  take  place  simultaneously,  but  for  our  purpose 
we  may  suppose  them  to  take  place  separately  ;  the  orbits 
becoming  circular  before  there  are  any  perceptible  changes 
in  the  rotations. 

0  Tfo  T>  Tfi 

Since  -^-.,  =  —  =  -~  ,  which  is  exceedingly  small,  we  may 

neglect  the  angular  momentum  of  translation  of  the  sun 
in  comparison  with  that  of  the  earth.  We  may  therefore 
treat  8,  the  centre  of  mass  of  the  sun,  as  a  fixed  point, 
and  E,  the  centre  of  mass  of  the  earth,  as  a  moving  point. 
Hence  so  long  as  the  rotations  are  supposed  unaffected,  or 
the  angular  momentum  of  translation  of  the  body  E  about 
S  constant,  we  may  apply  Kepler's  laws.  Thus  if  (a,  b) 
be  the  semi-axes  of  the  ellipse  described  by  E  about  S, 
e  its  eccentricity,  and  T  the  periodic  time,  or  length  of 
the  year,  Kepler's  third  law  gives 


But  if  -j-  be  the  rate  of  description  of  areas  about  S  by 

fit 

the  line  8E, 


dt 


206  ELEMENTARY   THERMODYNAMICS. 

fdA\* 
Hence  (-^  ) 


«  a  (1  -  O- 

Now  if  t>  be  the  velocity  of  the  point  E  about  S,  ds  an 
element  of  the  orbit,  and  p  the  perpendicular  from  S  on 
the  tangent,  we  have  evidently 


dA 
and  *PV==~ 


But  .Ejw  is  the  angular  momentum  of  translation  of  the 
body  E  about  the  fixed  point  S,  which  is  supposed  con- 
stant. Hence  the  value  of  a  (I—  e2)  remains  constant; 
and  therefore,  if  a'  be  the  radius  of  the  orbit  when  it  has 
become  circular, 

a'  =  a  (1  -  e-). 

Taking  the  present  value  of  e  to  be  ^,  we  see  that  when 
the  orbit  becomes  circular,  the  mean  distance  will  have 
diminished  by  the  3600th  part,  and  the  length  of  the 
year  by  the  2400th  part,  that  is,  by  3'65  hours. 

While  the  orbit  of  E  about  S  is  thus  becoming  circular, 
the  angular  velocity  of  translation  will  never  vary  much 
from  its  present  value.  Hence  when  the  circular  motion 
commences,  the  angular  velocity  of  translation  will  be 


CARNOT'S  PRINCIPLE.  207 

considerably  less  than  both  those  of  rotation.  Tidal 
friction  will  then  evidently  check  both  of  the  axial  rota- 
tions, drive  the  earth  further  away  from  the  sun,  and 
increase  the  length  of  the  year.  The  consequent  diminu- 
tions in  the  angular  momenta  of  rotation  will  be  attended 
by  an  exactly  equal  increase  in  the  sum  of  the  angular 
momenta  of  translation ;  and  the  mechanical  kinetic 
energies  of  rotation  and  translation  will  decrease,  partly 
to  supply  the  increase  in  the  mutual  potential  energy  of 
the  two  bodies,  and  partly  to  supply  the  non-mechanical 
kinetic  energy  developed  by  tidal  friction. 

In  making  a  rough  comparison  of  the  effects  of  tidal 
friction  on  the  angular  momenta  of  rotation  of  the  sun 
and  earth,  it  will  be  sufficient,  on  account  of  the  great 
distance  between  the  two  bodies,  to  treat  them  as  spheres 
of  radii  as,  ae. 

The  tides  are  due  to  the  fact  that  neither  body  attracts 
all  parts  of  the  other  alike.  Thus  the  attractions  of  the 
earth  on  masses  of  one  gramme,  situated  on  the  nearest 
and  remotest  parts  of  the  sun's  surface,  are  respectively 

r-   and  -r^- ,   the    difference   between 


which  is  roughly  —^  -  ^  .       8    .     Hence  since  each  of 

•v     « 

the  masses  is  attracted  by  the  sun  with  a  force  —  ,  we 

Q>8 

may  take  -~-  (i»-^—  )  as  a  measure  of  the  tide-producing 
force  on  the  sun.     Similarly,    „  (  p~~)    will  represent 


the  tide-producing  force  on  the  earth.     Consequently,  the 
tide-producing  force  on  the  earth  exceeds  that  on  the  sun 


208         ELEMENTARY  THERMODYNAMICS. 


in  the  ratio  ( -^)  (  —  i  .     Denoting  the  sun's  mean  density 
\J^J  \as/ 

by  ps,  and  the  earth's  by  pe,  this  ratio  is  equal  to  -^  -, 

or  about  94000.  J     We  therefore  conclude  that  the  tides 
will  be  much  higher  on  the  earth  than  on  the  sun. 
Again,  not  only  are  the  tides  much  smaller  on  the  sun 
than  on  the  earth,  but  the  force  of  gravity  is  greater  on 
the  surface  of  the  sun  than  on  that  of  the  earth  in  the 

ratio  f,  (  —  }  ,  or    *'"s    or  29.     It  may  therefore  be  inferred 
E  \aj         aepe 

that  if  the  sun  and  earth  be  removed  to  a  considerable 
distance  from  one  another  without  allowing  the  tides  to 
undergo  any  sensible  alterations,  and  the  two  bodies  be 
then  set  at  liberty,  the  tides  would  subside  much  more 
rapidly  on  the  sun  than  on  the  earth.  Thence  it  may  be 
assumed  that  when  the  two  bodies  are  rotating  freely 
with  equal  angular  velocities  within  the  range  of  one 
another's  influence,  the  tides  that  will  be  produced  will 
be  much  more  nearly  symmetrical  about  the  line  SE  in 
the  case  of  the  sun  than  in  the  case  of  the  earth. 
Thus  so  long  as  the  earth  revolves  on  its  axis  with  any 
moderate  speed,  tidal  friction  will  have  a  much  more 
energetic  effect  on  the  angular  momentum  of  the  earth's 
rotation  than  on  that  of  the  sun. 

In  general,  if  two  unequal  bodies  of  the  same  liquidity 
and  density  are  rotating  about  their  axes  with  the  same 
angular  velocity,  and  be  placed  in  presence  of  each  other, 
the  rotation  of  the  smaller  body  will  be  modified  by  tidal 
friction  much  faster  than  that  of  the  other,  because  the 

1  The  values  of  p,,  p,  are  calculated  from  the  data  given  in  Herschel's 
'  Astronomy.' 


CARNOT'S  PRINCIPLE.  209 

angular  momentum  of  rotation  is  less,  and  the  force 
which  tends  to  modify  it  greater,  than  in  the  case  of 
the  larger  body.  This  is  probably  the  reason  why  the 
moon  has  been  already  caused  to  turn  the  same  face  to 
the  earth  while  the  earth  does  not  yet  turn  the  same 
face  to  the  moon. 

Returning  to  the  case  of  the  sun  and  earth,  we  see  that, 
after  the  orbits  become  circular,  the  increase  in  the  sum 
of  the  angular  momenta  of  translation  will,  at  first,  be 
chiefly  due  to  the  decrease  of  the  diurnal  rotation  of  the 
earth.  This  will  go  on  until  the  earth  always  presents 
the  same  face  to  the  sun,  and  then  it  will  proceed,  at  a 
much  slower  rate,  at  the  expense  of  the  rotation  of  the 
sun. 

It  will,  of  course,  be  easy  to  calculate  what  will  be  the 
distance  between  the  earth  and  sun  when  they  show  the 
same  faces  to  one  another ;  but  we  may  also  estimate  the 
distance  when  the  earth  first  shows  the  same  face  to  the 
sun,  by  assuming  that,  until  that  time,  tidal  friction 
produces  no  effect  on  the  rotation  of  the  sun.  For  if  we 
take  the  earth's  '  radius  of  gyration '  to  be  a  third  of  her 
radius,  and  remember  that  the  ratio  of  r  to  R  +  r  is 
always  very  nearly  unity,  we  have  roughly,  when  the 
orbits  first  become  circular, 


=(  — 

V70000, 


49  x  108' 

Hence  the  ratio  of  the  angular  momentum  of  the  earth  to 
p.  14 


210  ELEMENTARY   THERMODYNAMICS. 

the  sum  of  the  angular  momenta  of  translation  of  the 
earth  and  sun,  will  at  first  be  about 

.365 
49  x  108' 

13xTO«- 

Now  since  the  orbital  angular  velocity  of  the  earth  about  the 
sun  decreases  as  the  circular  orbit  increases  and  is  already 
very  small,  it  is  clear  that  when  the  earth  is  caused  to  turn 
the  same  face  to  the  sun,  she  will  have  practically  lost  the 
whole  of  her  angular  momentum  of  rotation.  Hence  from 
the  time  that  the  orbits  first  become  circular  to  the  time 
when  the  earth  begins  to  turn  the  same  face  to  the  sun,, 
the  sum  of  the  angular  momenta  of  translation  will  have 

increased  by  its  y^  —  --7j6  ^n  Paft>  and  therefore,  by  equa- 

tion (49),  the  distance  between  the  two  bodies  will  have 

2 
increased  by  its  y^  —  —  th  part. 

Thus  the  sum  of  the  angular  momenta  of  translation  and 
the  distance  between  the  earth  and  sun,  will,  on  our 
suppositions,  be  scarcely  changed  until  tidal  friction 
begins  to  affect  the  rotation  of  the  sun. 

Again,  if  we  take  the  sun's  radius  of  gyration  to  be  a 
third  of  his  radius,  we  have,  at  present, 


which  is  rather  less  than  1. 

Thus  the  ratio  of  the  sun's  angular  momentum  of  rotation 

to  the  sum  of  the  angular  momenta  of  translation,  or 


CARNOT'S  PRINCIPLE.  211 

Hence  when  the  earth  and  sun  show  the  same  faces  to 
one  another,  the  sum  of  the  angular  momenta  of  transla- 
tion will  be  15  '6  times  as  large,  and  their  distance  243 
times  as  large,  as  they  are  now.  The  year  will  then  be 
3796  times  as  long  as  at  present. 

If  the  planet  E  had  been  mercury  instead  of  the  earth, 
the  last  result  would  have  been  quite  different.  For  we 
should  then  have  had,  at  present, 


. 

Er*     E  V24( 
=  84,  nearly. 

Consequently  the  ratio  of  the  sun's  angular  momentum  of 
rotation  to  the  sum  of  the  angular  momenta  of  translation 
of  the  sun  and  mercury,  is  at  present 

84  x88 
~25      ' 
or  about  295. 

Hence  when  the  two  bodies  are  caused,  by  tidal  friction, 
to  rotate  as  one,  the  distance  between  them,  on  the  sup- 
position that  all  other  bodies  are  absent,  will  be  87000 
times  as  great  as  it  is  now. 

In  comparing  the  changes  produced  by  tidal  friction 
in  the  mechanical  kinetic  energies,  after  the  orbits  have 
been  rendered  circular  and  before  E  turns  the  same  face 
to  S,  let  us  denote  corresponding  small  increments  by  the 
letter  d.  Then  d(%CEa)*)  =  CK<aEdo>E=  wEd(CEo>E\  and 
.  Thus 


Now  -  s  will  increase,  but  even  if  we  suppose  ws  to  remain 


14—2 


212  ELEMENTARY   THERMODYNAMICS. 

unchanged  until  the  earth  turns  the  same  face  to  the  sun, 

Q£l  * 

its  value  cannot  exceed  -^-  ,  or  14  '6.     On  the  other  hand, 
2o 

so  long  as  the  earth  possesses  any  considerable  angular 

rotation,  the  value  of  7      *    '\  will  be  exceedingly  small. 
d^fdt^) 

Hence  the  effects  of  tidal  friction  on  the  mechanical 
kinetic  energy  of  rotation  will  be  negligible  in  the  case 
of  the  sun  compared  with  that  of  the  earth. 
Again,  if  Ut  be  the  sum  of  the  mechanical  kinetic  energies 
of  translation  and  Ht  the  sum  of  the  angular  momenta  of 
translation,  equations  (49)  and  (50)  give 


so  that 


=  -<odHt, 

a  negative  quantity. 

But  so  long  as  we  can  neglect  the  changes  in  the  sun's 

angular  momentum  of  rotation, 


=  constant, 
or 
Hence 


Now  when  the  circular  orbit  undergoes  a  small  increase, 
the  increase  in  the  mutual  potential  energy  of  the  two 

bodies  isJ£+r)sdCR  +  r),aiid  the  decrease  in  the  sum 


CARNOT'S  PRINCIPLE.  213 

of  the  mechanical  kinetic  energies  just  half  as  much.  We 
therefore  see  that  the  whole  of  the  decrease  in  the  sum  of 
the  mechanical  kinetic  energies  of  translation,  and  the 

fraction  —   of  the   decrease   in   the   mechanical   kinetic 

WE 

energy  of  the  earth's  rotation,  will  be  required  to  supply 
the  increase  in  the  mutual  potential  energy,  and  that  only 

the  remaining  part,  1 ,  of  the  mechanical  kinetic  energy 

of  rotation  lost  by  the  earth  will  be  converted  into  non- 
mechanical  kinetic  energy  by  the  friction  of  the  tides. 

Since  —  will  have  its  least,  and  1 its  greatest  value, 

UE  0)f: 

just  after  the  orbits  become  circular,  it  follows  that  the 
non-mechanical  kinetic  energy  developed  by  tidal  friction 

364 

can  never  exceed  the  o^-th  of  the   mechanical  kinetic 
obo 

energy  lost  by  the  earth's  rotation. 

82.  The  effect  of  tidal  friction  in  retarding  the  rota- 
tion of  the  earth  about  its  axis,  that  is,  in  increasing  the 
length  of  the  day,  is  too  minute  to  be  detected  by 
instrumental  means ;  but  it  has  been  shown  to  exist  by 
the  calculations  of  Adams  and  Delaunay,  which  could  not 
be  made  to  agree  with  observation  when  the  length  of  the 
day  was  supposed  strictly  constant.  In  the  case  of  the 
other  planets,  their  periods  of  rotation  about  their  axes 
are,  at  present,  only  roughly  known.  We  cannot  therefore 
expect  to  have  any  independent  evidence  of  the  effects  of 
the  tides  on  their  speeds  of  rotation. 

Tidal  friction  affects  the  length  of  the  year,  partly  by 
making  the  orbit  circular,  and  partly  by  increasing  its 


214  EUEMEFTAiT  THEBHODTXAMIC&. 

dimensions.  From  the  former  cause,  the  year  may  be 
shortened  by  3  "65  Lours:  from  the  latter  it  will  be 
lengthened,  bin  the  increase  will  be  small  even  by  the 
time  xiit  earth  if  caused  to  nim  the  same  face  to  die 
sTm.  TpoL  uhe  -vrhok.  vr  conclude  that,  if  no  other  body 
were  near-  the  miL  and  earth.,  the  length  of  the  year,  owing 
to  tidal  friction,  would  be  slowly  shortening1. 

83  A«-  a  nearer  approximation  to  the  actual  etate 
of  the  solar  KvsteiL.  iet  u«  grjppose  that  there  are  two 
planet*  E.  J;  reToiring  about  «S'.  Then  if  we  can  neglect 
the  mutual  attraction  of  Z"  and  ./  in  comparison  with  the 
force*  vitt  winch  they  are  attracted  to  #,  it  is  clear  that 
if  E  and  J"  be  describing'  circular  orbits  of  different  sizes, 


- 

th^ar  period*  oi  j-ev  Nation  will  be  ^iifierent  It  will  there- 
lore  bfe  nupyjwibl*:  for  the  ^yistem  of  three  bodies  to  be 
luoring  m  rigid,  and  the  motion  will  consequently  be 
amended  bj  tidal  Mction,  In  this  case,  we  have  not 
inweJy  the  luuttiaJ  tidal  influence  of  A'  and  E  and  of  # 
mid  J,  but  of  if  and  J  a*  well  The  effects  produced  by 

H  >  flbimt-  it  iiit  yluwter    theyry  that,  on  the  supposition  that 


J  «i^    «».  „»  yatooy  Ti^  ^  dikturbmg  efleet.  of  the 

« 


then   dMraaung,   then   again 
»«ufc  10.0  n  ut,    2 


^  li*  Jy«rt  ^^cj-  rf  ^  airtnrf^  ^tj^.  of 
.fl  ti*  «iu 


CARNOT'S  PRINCIPLE.  215 

the  last-mentioned  tides  in  any  moderate  length  of  time 
will,  of  course,  be  quite  insignificant ;  but  if  sufficient 
time  be  allowed,  they  may  accumulate  to  a  considerable 
amount.  We  may  easily  obtain  a  rough  idea  of  these 
effects,  since  we  may  suppose  that  the  two  planets  have 
no  tidal  influence  on  one  another  except  when  they  are 
comparatively  near  together,  and  may  then  represent  the 
tides  produced  in  one  planet,  say  J,  by  the  other  planet 
E,  as  a  deformation  from  the  natural  spherical  shape  to  a 
slightly  prolate,  or  elongated,  spheroid  with  its  axis  point- 
ing roughly  to  E,  as  shown  on  an  exaggerated  scale  in  the 
figure. 

As  similar  tides  will  be  produced  in  E  by  J,  it  is  evident 
that  the  mutual  attraction  between  E  and  J  will  be 
slightly  greater  than  if  the  tides  were  completely  absent. 
The  effect  of  the  mutual  tidal  influence  of  E  and  /  will 
thus  be  to  draw  them  nearer  together,  or  to  diminish  the 
orbit  of  the  outer  planet  and  increase  that  of  the  inner. 

When  the  whole  solar  system  has  been  reduced,  by 
tidal  friction,  to  move  as  one  rigid  body,  let  any  number 

of  planets  E,  J, be  revolving  round  the  sun  in  such 

a  way  that  the  force  with  which  any  one  of  them  is 
attracted  to  the  sun  is  incomparably  greater  than  the 
other  forces  which  act  upon  it.  Then  if  o>  be  the  common 

angular  velocity  of  the  system,  and  rf,  rj( the  radii  of 

the  circular  orbits,  we  have 

— 2  =  reo),  or  S  =  re3to ; 


Hence 


216  ELEMENTARY  THERMODYNAMICS. 

If  two  planets  Ef,  J',  be  so  near  together  that,  in  the 
case  of  one  or  both  of  them,  their  mutual  attraction 
cannot  be  neglected  in  comparison  with  the  attraction  of 
the  sun,  it  will  be  evident  from  the  figures,  since  the 


J1. 


E' 


resultant  force  on  every  isolated  planet  must  pass  through 
the  centre  of  mass  of  the  whole  system,  which  is  practically 
the  same  as  the  centre  of  S,  that  the  two  planets  E',  J', 
must  lie  in  a  straight  line  with  S. 

It  will  now  be  clear  that  the  planets  will  ultimately 
form  a  ring  round  the  sun,  since  those  planets  which  are 
ultimately  at  a  considerable  distance  from  one  another, 
will  be  at  the  same  distance  from  the  sun,  and  since  those 
planets  which  are  behind  one  another  will  be  compara- 
tively near  together.  This  ring  will  resemble,  in  some 
respects,  those  which  now  surround  the  planet  Saturn ; 
the  chief  difference  being  that,  whereas  the  number  of 
small  bodies  of  which  Saturn's  ring  is  supposed  to  be 
formed,  must  be  very  great,  the  planets  which  will  con- 
stitute the  ring  round  the  sun  will  be  but  few. 

It  is  interesting  to  notice  that  we  can  conceive  of  no 
ultimate  arrangement  of  the  planets  which  would  admit 
of  these  small  bodies  continuing  to  form  circular  rings 
around  Saturn,  as  at  present.  We  therefore  conclude 
that  the  beautiful  rings  of  Saturn  will  be  broken  up.  For 
a  long  time  to  come,  however,  their  existence  will  probably 


CARNOT'S  PRINCIPLE.  217 

be  preserved,  in  spite  of  the  adverse  influences  of  the  rest 
of  the  solar  system,  by  the  friction  of  the  tides  due  to  the 
rapid  axial  rotation  of  Saturn. 

The  size  and  period  of  rotation  of  the  ring  which  will 
ultimately  encircle  the  sun  can  be  readily  calculated. 
For  it  can  easily  be  shown  that,  at  present,  the  sun 
contributes  less  than  2  per  cent,  of  the  total  angular 
momentum  of  the  solar  system  about  its  centre  of  mass. 
Hence  since  the  rotation  of  the  sun  about  his  axis  in  the 
final  state  of  the  system  will  be  vastly  smaller  than  now, 
we  see  that  the  ultimate  planetary  ring  will  contain 
practically  the  whole  of  the  angular  momentum.  Again, 
it  may  be  shown  that,  at  present,  Jupiter,  by  his  revolu- 
tion round  the  sun,  contributes  60  per  cent,  of  the  total 
angular  momentum  of  the  system,  while  his  mass  is  74 
per  cent,  of  the  sum  of  all  the  planetary  masses.  If 
therefore  the  ultimate  planetary  ring  were  coincident 
with  the  present  orbit  of  Jupiter,  the  total  angular 
momentum  of  the  solar  system  would  only  be  about  80 
per  cent,  of  what  it  is  now.  The  radius  of  the  ring  will 
therefore  be  about  (f)2  times  Jupiter's  present  distance 
from  the  sun,  that  is,  1200  million  kilometres,  or  750 
million  miles ;  and  its  period  of  rotation  will  be  about  (f  )3 
times  the  present  length  of  Jupiter's  year,  that  is,  nearly 
22  of  our  present  years. 

We  can  now  predict  the  fate  of  the  moon,  on  the 
assumption  that  no  other  planetary  bodies  ever  come  so 
near  the  earth  and  moon  as  to  exert  a  sensible  disturbing 
force  on  them.  For  if  S  be  the  mass  of  the  sun,  M  that 
of  the  moon,  rs  the  present  distance  of  the  earth  from  the 
sun,  and  rm  its  present  distance  from  the  moon,  the  tide- 
producing  force  on  the  earth  due  to  the  action  of  the  sun, 


218  ELEMENTARY   THERMODYNAMICS. 

is,  at   present,  to   the   tide-producing  force   due   to   the 

action  of  the  moon,  in  the  ratio  ,-,  (  —  )  ,  which  is  about  -£r. 
M\r,J 

Hence  it  is  clear  that  the  angular  momentum  lost  by  the 
earth's  rotation  goes  partly  to  enlarge  the  orbit  of  the 
earth  about  the  sun,  and  partly  to  enlarge  the  orbit  of  the 
moon  about  the  earth.  Even  if  the  whole  of  the  angular 
momentum  of  rotation  lost  by  the  earth  went  to  drive  the 
moon  further  away,  the  effect  would  not  be  considerable. 
For  if  E  be  the  mass  of  the  earth  ;  k  (=  £  the  radius)  the 
radius  of  gyration  ;  WF  the  present  angular  velocity  of 
the  earth's  axial  rotation  ;  and  &>m  the  present  angular 
velocity  of  the  moon's  revolution  about  the  earth  ;  the 
ratio  of  the  angular  momentum  of  rotation  of  the  earth 
to  the  moon's  revolutional  angular  momentum  about  the 
earth,  is,  at  present, 


180.8 


8 
109  ' 


Thus  even  on  the  supposition  that  the  sum  of  the  angular 
momenta  of  the  earth  and  moon  about  their  common 
centre  of  mass  remains  constant,  the  distance  of  the  moon 
from  the  earth  can  never  exceed  its  present  value  by  more 
than  the  T\f¥th  part  ;  and  the  length  of  the  month  can  not 
become  more  than  If  times  as  great  as  at  present,  that 
is,  can  not  exceed  33£  of  our  present  mean  solar  days. 
Since  if  the  earth  and  moon  continue  separate,  the  month 
and  the  year  must  ultimately  become  of  the  same  length, 
and  therefore  equal  to  21  or  22  of  our  present  years,  it  is 
clear  that  the  moon  must  fall  into  the  earth.  This  may 


CARNOT'S  PRINCIPLE.  219 

also  be  shown  as  follows.  Suppose,  if  possible,  that  the  sun, 
earth,  and  moon,  are  in  a  straight  line  with  one  another 
and  moving  like  a  single  rigid  body.  Then  if  the  distance 
of  the  moon  from  the  sun  be  1  +  x  times  the  distance 
of  the  earth  from  the  sun,  where  x  is  either  positive 
or  negative,  the  acceleration  of  the  moon  will  be  1  +  x 
times  the  acceleration  of  the  earth.  But  if  #  be  a  small 
quantity  and  P  the  attraction  of  the  sun  on  each  gramme 
of  the  earth,  the  attraction  of  the  sun  on  each  gramme  of 

p 
the  moon  will  be  ,     —  ^-,  or  P  (1  —  2#).     Hence,  since  we 

may  neglect  the  attraction  of  the  moon  on  each  gramme 
of  the  earth  in  comparison  with  the  attraction  of  the 
earth  on  each  gramme  of  the  moon,  the  attraction  of  the 
earth  on  each  gramme  of  the  moon  must  be  ±  3Px,  or  the 
earth's  mass  +  3^  times  that  of  the  sun.  Thus 


Since  the  328th  part  of  the  ultimate  distance  of  the 
earth  from  the  sun  is  several  times  greater  than  the 
greatest  possible  distance  of  the  moon  from  the  earth,  we 
conclude,  as  before,  that  the  moon  cannot  remain  perma- 
nently separate  from  the  earth. 

It  is  easy  to  see  how  the  moon  and  earth  will  be 
brought  together.  For  a  long  time  to  come,  the  tides 
will  have  the  effect  of  driving  the  moon  further  away 
from  the  earth.  This  will  go  on  until  the  day  is  nearly 
as  long  as  the  month,  and  then  the  lunar  tides  will 
practically  cease  and  the  distance  of  the  moon  become 


220  ELEMENTARY  THERMODYNAMICS. 

stationary.  Owing  to  the  continued  action  of  the  solar 
tides,  the  day  will  at  length  become  so  much  longer  than 
the  month  that  lunar  tides  will  again  be  called  into 
existence  and  the  moon  begin  to  approach  the  earth. 
The  action  of  the  tides  will  then  transfer  angular  momen- 
tum from  the  moon's  revolution  about  the  earth  to  the 
earth's  axial  rotation,  and  from  the  latter  to  the  earth's 
orbit  about  the  sun.  As  the  moon's  orbit  about  the  earth 
contracts,  the  two  bodies  will  begin  to  deviate  from  mov- 
ing as  rigid,  and  they  will  at  last  come  into  violent 
collision.  This  conclusion  will  be  made  clear  by  the 
following  argument.  As  the  two  bodies  approach  one 
another,  let  them  be  supposed  to  move  as  one  rigid  body. 
Then  if  H  be  the  sum  of  their  angular  momenta  about 
their  common  centre  of  mass,  and  r  the  distance  between 
them,  we  have,  by  equations  (48)  and  (49), 


or,  nearly,         H  =  M  Vxj£r   +  L  , 

r% 

where  the  roots  are  to  be  taken  positively. 
Hence          iff. 


Thus  if  dH  and  dr  are  both  to  be  negative,  /•  will  have  to 
be  greater  than  W264,  that  is,  greater  than  16'2&.  Con- 
sequently if  we  take  k  to  be  J  of  the  earth's  radius,  the 
least  possible  value  of  r  will  be  5  '4  times  the  earth's 
radius,  that  is,  34,500  kilometres,  or  21,400  miles.  When 


CARNOT'S  PRINCIPLE.  221 

this  value  of  r  is  reached,  the  motion  which  we  have 
assumed  for  the  system  ceases  to  be  possible. 

The  only  other  planet  we  shall  here  consider  is  Uranus, 
whose  satellites  are  the  only  bodies  in  the  solar  system 
with  the  exception  of  comets  which  are  known  to  possess 
a  retrograde  motion.  In  this  case,  by  supposing  the  tides 
produced  by  the  satellites  and  by  the  sun  to  act  alternately 
for  equal  short  times,  it  will  easily  be  seen  that  the 
satellites  will  be  continually  drawn  nearer  the  primary, 
whose  axial  rotation,  though  it  will  be  checked,  cannot 
be  rendered  retrograde.  Finally,  the  satellites  will  all 
fall  into  the  primary  and  the  whole  motion  will  become 
direct. 

84.  Another  effect  of  friction  worthy  of  notice  is  that 
due  to  the  action  of  running  Avater  and  glaciers ;  but 
between  these  agencies  and  tidal  friction  there  is  an 
important  difference,  that,  whereas  the  tides  are  merely 
due  to  want  of  'perfect  rigidity,'  and  may  exist  in  any 
body,  whether  solid,  liquid,  or  gaseous,  the  effects  of 
running  water  and  glaciers  will  rapidly  diminish  and  dis- 
appear when,  owing  to  the  cooling  down  of  the  sun,  the 
seas  become  frozen  up,  and  rain  and  snow  cease  to  fall. 

Running  water  and  glaciers  are  continually  at  work 
wearing  away  the  land  and  transporting  the  detritus  to 
lower  places  or  to  the  sea,  where  fresh  land  is  gradually 
forming.  If  this  action  should  last  long  enough,  without 
anything  to  counteract  it,  the  land  would,  in  time,  be 
everywhere  reduced  to  the  level  of  the  sea.  However, 
there  are  two  agents  in  operation  which  have  a  modifying 
influence.  In  the  first  place,  the  earth's  crust  is  liable  to 
be  upheaved  by  volcanic  and  similar  forces ;  and  secondly, 


222 


ELEMENTARY   THERMODYNAMICS. 


the  slackening  of  the  earth's  rotation  about  its  axis, 
owing  to  tidal  friction,  enables  the  waters  of  the  sea,  so 
long  as  they  remain  unfrozen,  to  slowly  flow  from  the 
equator  towards  the  poles,  and  so  tends  to  submerge  the 
land  about  the  poles,  and  to  make  it  more  elevated  in  the 
neighbourhood  of  the  equator.  Now  on  referring  to  works 
on  geology,  it  will  be  seen  that  the  denuding  effects  of 
running  water  are  amply  sufficient  to  neutralize  the 
tendency  of  tidal  friction  to  cause  an  equatorial  elevation 
of  the  land.  We  therefore  see  that,  if  it  were  not  for 
volcanic  and  similar  agencies,  the  earth  would  acquire  a 
more  even  surface,  with  an  increased  quantity  of  land  in 
the  equatorial  regions ;  and  that,  in  the  polar  regions,  the 
land  would,  in  some  places,  be  submerged  by  the  rising  of 
the  sea,  and  in  others,  new  land  would  be  formed  by  the 
deposits  of  rivers  and  glaciers. 

85.     The    effect   of    tidal   friction  in  shortening  the 
periodic  time  by  making  the  orbit  more  circular,  may  be 


expected  to  be  most  considerable  in  the  case  of  very 
elongated   elliptical   orbits,   like   those   of  comets.     For, 


CARNOT'S  PRINCIPLE.  223 

since  by  Kepler's  second  law,  the  line  joining  the  comet  to 
the  sun  describes  equal  areas  in  equal  times,  it  is  obvious 
that  the  comet's  orbital  angular  velocity  will  be  so  ex- 
tremely small  in  the  more  distant  parts  of  the  orbit  that 
almost  the  whole  of  the  periodic  time  will  be  spent  in 
traversing  these  parts.  Hence  tidal  friction,  by  merely 
making  the  orbit  slightly  less  elongated,  and  consequently 
diminishing  those  parts  of  it  which  require  the  most  time 
for  their  description,  may  produce  a  sensible  diminution  in 
the  periodic  time. 

A  change  in  the  periodic  time  has  only  yet  been 
observed  in  the  case  of  Encke's  comet,  and  the  observed 
fact  is  a  shortening  of  the  period.  This  may  be  due, 
partly  to  tidal  friction,  and  partly  to  solar  radiation,  as 
explained  in  the  foot-note1;  but  a  different  explanation, 
propounded  by  Encke  himself,  has  hitherto  been  almost 
universally  accepted,  at  least,  until  quite  recently.  It 
was  supposed  that  the  ether  offered  a  slight  resistance  to 
the  mechanical  motion  of  the  comet  through  space,  in- 
dependent of  the  non-mechanical  motions  (see  Art.  59), 
from  which  it  would  follow  that  those  parts  of  the  orbit 
which  require  most  time  for  their  description,  would  be 
cut  shorter  every  revolution,  just  as  by  tidal  friction ;  and 
in  consequence,  that  the  periodic  time  would  diminish. 

1  When  a  comet  approaches  the  sun,  it  is  generally  observed  to  throw 
off  a  tail,  which  points  away  from  the  sun,  and  is  not  merely  left  behind 
by  the  comet.  The  cause  of  the  phenomenon  is  evidently  equivalent  to  a 
repulsive  force  residing  in  the  sun.  Such  a  force  will  have  no  effect  on 
the  angular  momenta  of  the  comet ;  but,  since  the  tail  only  appears  when 
the  comet  is  in  that  part  of  its  orbit  nearest  the  sun,  it  will  make  the  orbit 
more  circular,  just  like  tidal  friction,  but  perhaps  more  energetically. 

We  have  already  stated  how  we  consider  it  probable  that  the  tails  of 
comets  are  produced  by  solar  radiation. 


224  ELEMENTARY   THERMODYNAMICS. 

The  mere  fact  that  the  periodic  time  of  a  single  comet 
is  undergoing  a  slow  diminution,  appears  to  be  a  slender 
foundation  for  Encke's  hypothesis  of  an  etherial  resistance 
to  mechanical  motion,  and  accordingly,  there  has  lately 
been  a  disposition  to  reject  it1.  Now  that  other  causes 
have  been  shown  capable  of  explaining  the  comet's  peculiar 
behaviour,  we  must  admit  that  the  evidence  in  favour  of 
Encke's  theory  is  very  slight  indeed2 ;  and  for  this  reason, 
it  has  not  hitherto  been  taken  account  of  in  this  work. 
Still,  as  it  has  not  been  completely  disproved,  it  is  necessary 
to  allow  the  possibility  of  it,  and  to  examine  what  would 
be  its  effects  on  the  state  of  the  solar  system.  This 
question  is  interesting  in  itself  and  historically  important, 
partly  from  the  frequent  references  that  are  made  to  it, 
and  also  because  Encke's  theory  was  universally  accepted 
at  the  time  that  the  final  state  of  the  solar  system  was 
predicted  by  Sir  W.  Thomson. 

If  we  assume  that  the  only  external  influences  to 
which  the  solar  system  is  subject  are  the  resistance  of 
the  ether  and  the  radiation  of  energy  into  space,  then,  in 
the  final  state,  every  part  will  be  at  the  temperature 
of  absolute  zero,  there  will  be  no  mechanical  motions, 
and  the  different  bodies  will  all  lie  together,  instead  of 

1  Newcomb's  '  Popular  Astronomy.' 

2  A  theoretical  argument  which  is  sometimes  thought  to   support 
Encke's  hypothesis,  is  given  in  the  late  Prof.  Balfour  Stewart's  '  Treatise 
on  Heat.'    By  assuming  that  the  laws  of  radiation  are  exactly  the  same 
for  bodies  in  mechanical  motion  as  for  bodies  in  a  state  of  mechanical 
rest,  it  is  concluded  that  the  principle  of  energy  and  Camot's  principle 
cannot  both  be  true  unless  there  be  an  etherial  resistance  to  mechanical 
motion.     The  obvious  reply  to  this  argument  is  that  the  laws  of  radiation 
cannot  be  quite  the  same  for  bodies  in  rapid  mechanical  motion  as  for 
bodies  in  a  state  of  mechanical  rest. 


CARNOT'S  PRINCIPLE.  225 

being  separated  by  considerable  distances,  as  at  present. 
Consequently,  in  the  final  state,  there  will  be  neither 
mechanical  nor  non-mechanical,  kinetic  energy  of  matter  ; 
but,  owing  to  the  force  of  gravitation,  there  will  be 
potential  energy,  that  is,  ethereal  kinetic  energy  bound 
to  the  material  system. 

Again,  if  we  take  any  number  of  spherical  bodies,  the 
mass  M,  of  any  one  of  them  whose  radius  is  r  and  density 

p,  will  be  0  vrr3/?  ;  while  the  area  A,  which  will  be  exposed 
o 

to  the  effects  of  a  resisting  medium,  will  vary  as  r2.  We 
therefore  have  A=k(—},  where  k  has  the  same  value  for 


all  the  bodies.  Hence,  if  we  make  the  usual  assumption 
that,  for  a  given  velocity  of  the  centre  of  mass,  the 
resistance  oc  A,  it  is  clear  that  the  time  required  by  the 
resistance  to  diminish  a  given  velocity  by  a  given  amount, 
will  X  (Mp-)*- 

Now  the  sun  and  planets  immensely  exceed  a  comet  in 
mass,  and  are  probably  not  inferior  in  density.  The 
resistance  of  the  ether  will  therefore  have  a  vastly  more 
important  influence  on  the  motion  of  a  comet  than  on 
that  of  the  sun  or  a  planet.  But,  according  to  Encke's 
theory,  the  effect  of  the  resistance  is  small,  even  in  the 
case  of  a  comet.  Hence  in  the  case  of  the  sun  and  larger 
planets,  it  will  be  so  insignificant  that  it  may  be  supposed 
not  to  come  into  operation  so  long  as  there  is  any  appre- 
ciable amount  of  tidal  friction  at  work,  and  at  all  times 
the  mechanical  motions  relative  to  the  centre  of  mass  will 
be  sensibly  the  same  as  if  that  point  were  at  rest. 
As  soon  as  the  whole  system  begins  to  move  practically 
as  a  rigid  body,  the  resistance  of  the  ether  will  come  into 
P.  15 


226  ELEMENTARY   THERMODYNAMICS. 

action  and  cause  the  different  bodies  to  describe  nearly 
circular  spirals.  The  motion  will  continue  to  be  of  this 
character  until  a  collision  occurs,  or  one  of  the  orbits 
becomes  unstable,  when  tidal  friction  will  again  become  of 
overwhelming  importance 

86.  One  of  the  most  important  and  interesting  parts 
of  the  solar  system  problem  is  the  question  of  the  origin  of 
the  energy  which  is  radiated  so  copiously  by  the  sun. 
The  first  idea  which  suggests  itself  to  us,  is  that  the  sun's 
'  heat '  is  due  to  chemical  action  ;  but  this  will  be  shown 
to  be  inadmissible  by  considering  the  most  powerful  chemi- 
cal action  known,  viz.,  the  union  of  oxygen  and  hydrogen 
to  form  water. 

If  we  take  8  grammes  of  oxygen  and  one  gramme  of 
hydrogen  at  0°  C.  and  a  pressure  of  one  atmo,  and  by  first 
imparting  and  afterwards  abstracting  heat,  cause  them  to 
form  9  grammes  of  water  at  0"  C.  under  a  uniform  normal 
pressure  which  remains  constantly  equal  to  one  atmo 
during  the  process,  the  heat  evolved  will  be  about  34,000 

34,000 
calories,  or  — - —  calories  per  gramme  of  water  formed. 

Now  if  we  treat  the  sun  as  a  perfect  sphere  of  r  centi- 
metres radius  and  uniform  density  p,  and  suppose  that  the 

total  amount  of  heat  which  it  can  radiate  is  — calories 

t/ 

per  gramme,  or  -  -rrpr3  x  — -g -—  calories  in  all ;  the  total 

amount  of  heat  radiated  from  each  square  centimetre  of 
the  surface,  will  be 

1          34,000 

s  pr  X  —     —  calories . 

o  y 


CARNOT'S  PRINCIPLE.  227 

Taking  r  to  be  71  x  109,  and  p,  -,  the  last  expression 
becomes 

5  x  71  x  109     34,000 

— ;; x  — - —  calories, 

4x3  9 

or  1 1 20  x  1011  calories. 

Hence  since  the  radiation  from  each  square  centimetre  of 
the  sun's  surface  is  1215  calories  per  second,  chemical 
action  can  only  keep  it  up  at  its  present  rate  for 

1120xlOn 


1215 

or  about  2900  years. 

Chemical  action  being  thus  quite  insufficient  to  account 
for  the  sun's  '  heat,'  we  are  driven  to  adopt  the  nebular 
hypothesis,  as  developed  by  Mayer,  Helmholtz,  and  Thom- 
son. According  to  this  theory,  the  sun  was  originally  in 
the  form  of  a  very  attenuated  gas,  in  which  state  it  would 
possess  an  enormous  amount  of  gravitational  potential 
energy ;  and  during  the  condensation  of  the  materials  to 
their  present  bulk,  the  original  potential  energy  has  been 
gradually  drawn  upon  to  supply  the  heat  which  has  been 
so  freely  radiated  away  for  countless  ages.  This  will  be 
made  clear  by  the  following  calculations. 

If  two  small  bodies  of  masses  m,  m' ,  change  their 
distance  apart  from  R  to  r,  the  work  done  upon  them  by 
their  mutual  gravitational  attraction,  or  the  decrease  in 
their  mutual  gravitational  potential  energy,  will  be 

[r     mm' ,      ,  .  ,  /I       1 

A,—  —  (-dr)  =  \mm( -= 

JE       ?  \T     R 

If  the  masses  be  originally  at  a  great  distance  from  one 

15—2 


228  ELEMENTARY    THERMODYNAMICS. 

another,  we  may  put  R  infinite,  and  the  integral  becomes 
simply 


mm 
r 

Hence   the   gravitational   potential    energy   lost   by   the 
materials  of  the  sun  during  condensation  may  be  written 

xS7^, 

r 

where  2  refers  to  the  present  state. 

Denoting  the  distance  between  the  two  small  masses  mp, 
mq,  by  rpq,  we  have 
^>  mm' 

A.2,  -  =  A, 


l-x       i2  ,      i*     m1mi 

2  ^  ")~  r 


Thus  if  we  put 

~n»      ria  •  rlt  ' 

^21      r-a      rM 


CARNOT'S  PRINCIPLE. 


we  obtain 


229 


We  may  therefore  determine  the  value  of  XS  ----  from 

the  properties  of  the  simple  function  F,  which  is  such 
that  its  value  at  any  point  P,  called  the  potential  of  the 

system  at  P,  is  S  —  ,  where  r  is  the  distance  of  the  small 

mass  TO  from  P,  and  the  summation  extends  to  the  whole 
material  system  with  the  exception  of  any  small  masses 
which  may  be  situated  indefinitely  near  the  point  P. 

87.  Take  a  thin  homogeneous  spherical  shell  of 
density  p,  radius  a,  and  thickness  r,  and  let  us  find  its 
potential  at  an  internal  point  P. 

Draw  any  line  TPT'  through  P  meeting  the  shell  in 
T,  T',  and  join  T  and  T  to  the  centre  C,  With  P  as 


vertex  and  TPT'  as  axis,  describe  a  small  cone  of  vertical 
solid  angle  dto.  Then  if  PT=  r,  the  potential  at  P  of  the 
element  cut  by  the  cone  from  the  shell  at  T,  will  be 

r-rpda)     1  _    rrpdw 
cosCTT"r~cosCTT" 


230  ELEMENTARY   THERMODYNAMICS. 

Similarly,  if  PT  =  r,  the  potential  at  P  of  the  element 
cut  from  the  shell  at  T',  is 

r'rpdco 
cosCT'T' 

Since  the  angles  CTT',  CT'T,  are  equal,  the  sum  of  these 
two  elementary  terms  is 

(r  +  ?•')  rpdw 

cos  err  ' 

or  2ciTpd(o. 

Hence  the  potential  at  P  of  the  whole  shell  is  ^jrarp, 
which  is  independent  of  the  position  of  P  and  the  same 
as  the  potential  at  the  centre  C. 

To  find  the  potential  at  any  external  point  Q,  take  an 
internal  point  P  on  CQ  such  that 


CP       a 
^  =  CQ> 
if  any  line  TPT'  be  drawn  through  P  meeting  the  shell 


in  T,  T,  the  triangles  OPT,  CTQ,  will  be  similar,  and 
therefore 

CQ      CT     a 


CARNOT'S  PRINCIPLE.  231 

Hence  since  the  potential  at  Q  of  the  element  at  T  is 

r-rpdw       1   _      rrpdw       a 

cos  CTT'  'TQ^cos  CTT  '  CQ  ' 

the   potential   of  the  whole  shell   at  Q  is  equal  to  its 

potential  at  P  multiplied  by  ^  ,  and  is  therefore  —  nn  °  , 
uy  G(j/ 

which  is  the  same  as  if  the  whole  shell  had  been  concen- 
trated into  a  particle  at  the  centre  G. 

Now  take  a  solid  homogeneous  shell  of  radius  a  and 
density  p,  and  let  us  find  its  potential  at  an  internal  point 
P,  which  is  at  a  distance  x  from  the  centre  G.  For  this 
purpose,  consider  an  elementary  spherical  shell  of  radius  r 
and  thickness  dr,  with  its  centre  at  G.  If  r  be  greater 
than  x,  the  point  P  will  be  inside  the  shell  and  the 

potential  of  the  shell  at  P  will  be  —  -  --  ,  or  4<Trprdr. 
If  r  be  less  than  a?,  the  point  P  will  be  outside  the  shell 
and  the  potential  of  the  shell  at  P  will  be  —  £—  —  . 
Hence  the  potential  at  P  of  the  whole  sphere  is 

2 

~  TT/MT. 

Q  « 

Since  when  we  put  a  and  x  both  zero,  the  last  expres- 
sion vanishes,  we  see  that  the  potential  at  any  point  P  of 
a  small  mass  surrounding  that  point  is  zero.  We  may 
therefore  remove  the  restriction  that  in  finding  the  poten- 
tial of  a  system  at  any  point  P,  we  are  to  take  no  account 
of  the  mass  situated  at  that  point. 

88.  In  applying  these  properties  of  the  potential,  we 
shall  treat  the  sun  as  a  perfect  sphere  of  radius  a  and 
uniform  density  p.  This,  of  course,  cannot  be  accurate, 


[a  .         ,        [ 
4arardr  -M 

Jx  JQ 


232  ELEMENTARY  THERMODYNAMICS. 

but  it  will  simplify  the  calculations  and  will  show  the 
value  of  the  nebular  hypothesis. 
With  this  assumption 


4 

If  jl/  be  the  whole  mass,  M=»  -rrpa3,  and  therefore 
o 

*x2m7=?X  — . 

D       a 

Hence,  taking  the  first  form,  and  putting  X  =  — ,  7r2  =  10, 
p  =  -  ,  a  =  7  x  1010,  we  find 


_  65  x  5  x  75  x  1042 
3 

=  18  x  1047. 
If  a  contract  from  its  present  value  to  a  (1  -  ,AAAA),  we 

\          lU,UUw 

have,  by  the  second  formula  for  £A,2 


=  -  18  x  104:i. 

Since  the  radiation  emitted  by  the  sun  in  a  year  is 
1041  ergs,  it  follows  that  in  the  contraction  of  the  sun  to 
its  present  size,  the  work  done  by  the  mutual  gravitation 
of  its  parts  would,  on  our  hypothesis,  supply  the  energy 


CARNOT'S  PRINCIPLE.  233 

radiated  at  the  present  rate  in  18  million  years,  and  a 
further  contraction  of  one  10,000th  part  of  the  diameter 
would  keep  up  the  present  rate  for  1800  years. 

89.  The  radiation  forces  at  the  surface  of  the  sun  can 
easily  be  estimated.  For  if  we  take  the  temperature  of 
the  sun's  surface  to  be  10,000°  C.,  the  average  non- 
mechanical  kinetic  energy  of  each  surface  particle  may 

be  taken  to  be       *       ,  or  37 1,  times  as  great  as  it  would 

be  at  0°  C.  We  may  therefore  assume  the  average  non- 
mechanical  velocity  of  each  surface  particle  at  6  x  50,000 
centimetres  per  second.  Hence,  since  the  radiation  from 
each  square  centimetre  of  the  sun's  surface  is  525  x  108 
ergs  per  second,  the  total  radiation  force  per  square 
centimetre  will  be  175,000  dynes. 

The  resultant  of  the  radiation  forces  acting  on  any 
finite  area,  found  according  to  the  usual  rules  of  statics, 
will  generally  be  quite  insignificant;  but  since  Carnot's 
principle  does  not  require  it  to  be  strictly  zero  except  in 
the  invariable  state,  it  is  necessary  to  suppose  that  the 
radiation  forces  can  have  a  moment  about  an  axis  through 
the  centre  of  mass.  As  a  basis  for  calculation,  let  us 
suppose  that  at  any  instant,  all  those  particles  which  are 
sending  out  radiation,  are  moving  in  such  a  way  that  the 
?i-th  part  of  the  radiation  forces  concur  to  oppose  the 
angular  rotation  of  the  sun.  Then  the  moment  about  the 
axis  of  rotation  of  the  radiation  forces  will  be 


2  f  175, 
nj  o 


1 75,000  x27ra3  sin2 
o 

175,000  x 


234  ELEMENTARY   THERMODYNAMICS. 

Now  if  to  be  the  angular  velocity  of  the  body  about  its 
axis,   the   angular   momentum    about   the   axis   may   be 


written  Cw,  or  J\Ik"o),  where  k  is  the  radius  of  gyration. 

,r7orfo>         iTSjOOO-Tr'a3 

Hence  Mk- -----  = — , 

dt  n 

4> 

or,  putting  -  jrpa3  for  M, 

,dw         525 


Consequently,  if  we  assume  k  =  ~  and  put  -  for  p,  we  find 

f  =  -945xlO»x— . 

dt  no? 

If  this  state  continue  for  a  finite  time, 

o>  =  e»0-945  xlO3  x  —  t, 
no? 

where  a>0  is  the  value  of  &>  when  t  =  0. 

Thus  <u  will  become  zero  in     .  _ — ~ seconds. 


CARNOT'S  PRINCIPLE.  235 

Substituting  for  a  and  &>0,  we  find 


945  x  103  x  TT 

=  n  x  7-  x  10-°  x  ^-.         27r 


25  x  24  x  (60)2     945  x  10s  x  TT 
1013  x  n 
58  x  36 
1011 


Reducing  to  years, 

1011  1011  x  n 

X  n  sec°nds  = 


•21  2Tx 

10U  x  n 


756  x  10s  x  876 
108  x  n 


ye£ 


66256 
=  151  x  n  years. 

From  this  it  appears  that  if  the  sun's  angular  momentum 
has  been  acquired  by  means  of  the  radiation  forces,  it 
must  have  taken  a  great  length  of  time,  since  we  cannot 
suppose  more  than  a  small  part  of  the  radiation  forces 
ever  to  have  concurred  to  give  a  mechanical  resultant. 

It  will,  of  course,  be  seen  that  we  cannot  accept 
Newton's  third  law  of  motion,  as  usually  stated,  that  to 
every  force  acting  on  a  material  particle  there  is  an 
exactly  equal  and  opposite  force  acting  simultaneously  on 
some  other  material  particle,  near  or  distant.  Throughout 
this  work  we  have  assumed  that  action  and  reaction  are 
equal  and  opposite  in  the  case  of  two  particles  in  contact, 
whether  these  particles  are  both  material,  or  both  ethereal, 
or  one  of  them  material  and  the  other  ethereal. 


236  ELEMENTARY   THERMODYNAMICS. 

90.  The  sun's  mechanical  kinetic  energy  of  rotation 
is  i(7<u2,  or  £  ^-"r" .  Thus  if  we  suppose  that  when  the 

L> 

sun  was  in  the  form  of  a  very  attenuated  gas  and  C 
consequently  very  great,  the  angular  momentum  of  rota- 
tion was  the  same  as  now,  the  mechanical  kinetic  of 
rotation  would  then  be  practically  zero.  Hence  if  we 
take  no  notice  of  the  translation  of  the  centre  of  mass 
and  suppose  the  absolute  temperature  to  have  been 
originally  zero,  we  may  give  the  following  brief  descrip- 
tion of  the  history  of  the  sun  within  the  period  over  which 
our  present  scientific  knowledge  extends. 

Originally,  the  energy  of  the  sun  was  chiefly  gravita- 
tional potential  energy,  or  ethereal  kinetic  energy  bound 
to,  or  entangled  among,  its  material  particles.  Then  by 
the  condensation  of  the  mass,  much  of  this  ethereal  kinetic 
energy  has  been  transferred  from  the  ether  to  the  material 
particles  of  the  sun,  and  by  the  latter  radiated  into  space, 
that  is,  set  free  or  lost. 

If  the  fixed  stars  have  no  disturbing  effect  on  the  sun,  an 
invariable  state  will  ultimately  be  attained  in  which  there 
will  be  no  non-mechanical  kinetic  energy  of  matter,  but 
in  which  there  will  be  some  gravitational  potential  energy, 
and  also,  unless  Encke's  hypothesis  be  true,  mechanical 
kinetic  energy  of  matter. 


CHAPTER  IV. 

APPLICATIONS  OF   CARNOT'S   PRINCIPLE. 

91.  IF  a  material  system  be  protected  from  external 
electric  influences,  the  principle  of  the  Conservation  of 
Energy  may  be  expressed  in  the  form 


and  if  at  every  instant  the  temperature  be  uniform 
throughout  the  system  and  the  operation  reversible, 
Carnot's  principle  gives 

dQ  =  0d(f>. 

These  two  important  fundamental  equations  will  be 
used  to  investigate  some  of  the  properties  of  bodies  of 
uniform  temperature  in  which  there  are  no  electric  actions 
and  no  mechanical  motions.  In  these  bodies  we  shall 
suppose  that  the  only  external  forces  are  gravity  and 
surface  pressures  ;  and,  except  when  it  is  expressly  stated 
otherwise,  only  those  changes  of  state  will  be  considered 
in  which  no  mechanical  work  is  done  on  the  body  except 
by  a  uniform  normal  surface  pressure.  Denoting  this 
pressure  by  p  and  the  volume  of  the  body  by  v,  we  have 

dW^-pdv, 


238  ELEMENTARY   THERMODYNAMICS. 

and  if  the  operation  be  reversible, 

dU=6d(f>-pdv. 

When,  as  frequently  happens,  any  two  of  the  five  quanti- 
ties (6,  p,  v,  U,  (/>)  can  be  taken  as  independent  variables 
to  define  the  state  of  the  body,  we  obtain  also 

d  ( U  +  pv)  =  6d<f>  +  vdp, 
and 


Hence  by  expressing  that  these  quantities  are  complete 
differentials,  we  get 

dt8  = 

dv 


de(f> 
dv 


These   results   are   known   as   'the   four   thermodynamic 
relations.' 

92.  The  problems  we  propose  to  discuss  in  the 
present  chapter  may  be  divided  into  three  classes.  In 
the  first,  the  body  is  supposed  to  be  homogeneous  through- 
out, like  a  gas  or  a  piece  of  iron  or  india-rubber.  In  the 
second,  it  is  supposed  to  consist  of  two  or  more  homo- 
geneous parts,  which  are  alike  in  substance  but  differ 
from  one  another  in  physical  state,  as  in  the  case  of  water 
and  steam,  or  ice  and  water.  In  the  third,  we  shall 
chiefly  consider  bodies  consisting  of  two  or  more  homo- 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE.  239 

geneous  parts  which  are  not  alike  in  substance.  As  an 
illustrative  example,  we  may  take  the  case  of  an  aqueous 
solution  of  a  salt  with  a  quantity  of  salt  undissolved. 


PART  I. 

ON   HOMOGENEOUS   BODIES. 

93.  We  shall  first  suppose  that  when  the  substance 
is  in  a  state  of  equilibrium,  we  may  take  any  two  of  the 
three  quantities  (0,  p,  v)  as  independent  variables,  and 
treat  the  third  as  a  dependent  variable.  Also,  for  sim- 
plicity, we  shall  take  the  mass  of  the  body  to  be  one 
gramme. 
Then  if  (6,  v)  be  chosen  as  independent  variables, 


dv 

But  if  Cvdd  be  the  heat  required  to  raise  the  temperature 
of  the  body  from  0  to  6  +  dd,  while  the  volume  is  kept 
constant,  we  have 


and  therefore 


-~dv  ...............  (53). 


We  shall  define  Cv  to  be  the  'specific  heat  of  the  substance 
at  constant  volume.'  This  definition  is  not  quite  the 
same  as  that  given  in  Art.  33  ;  but,  as  a  matter  of  fact, 
the  specific  heat  varies  so  little  with  the  temperature 
that  the  two  definitions  are  identical  for  experimental 
purposes. 


240  ELEMENTARY  THERMODYNAMICS. 

Again,  we  have 

dU=dQ-pdv. 

Hence 


which  may  be  writtten 

dO  =  CvdO  +  dj®  dv,  \ 
dv 

so  that 

d*Q  =  d»U  + 

dv        dv 

This  expression  for  dQ  only  holds  for  a  reversible  modifi- 
cation, because  we  have  supposed  the  body  to  be  always 
in  a  state  of  equilibrium. 

To  find  -p,  let  the  substance  be  made  to  undergo  the 
dv 

following  reversible  cycle  of  operations. 

(1)  Let  the  volume  be  slowly  increased  from  v  to 
v  +  dv,  whilst  the  temperature  is  kept  constantly  equal 

to  6.     The  heat  absorbed  will  be  —. 9,    dv. 

dv 

(2)  Let  the  state  of  the  substance  then  be  altered 
without  loss  or  gain  of  heat  so  that  the  temperature  falls 
to  6  —  T,  where  r  is  indefinitely  small. 

(3)  Let  the  volume  now  be  slowly  diminished  at  the 
constant   temperature  6  —  r  by  such  an   amount  that  in 
the  fourth  operation  in  which  there  is  neither  loss  nor 
gain  of  heat,  it  may  be  possible  to  restore  the  substance 
to  its  original  state. 

On  the  indicator  diagram,  the  cycle  will  be  represented 
by  a  small  parallelogram  A  BCD,  whose  area  will  be  equal 


APPLICATIONS   OF   CARNOT S   PRINCIPLE. 


241 


to  that  of  ABba,  where  Aa,  Bb  are  drawn  parallel  to  Op. 
Hence,  since  the  abscissae  of  A  and  B  are  v  and  v  +  dv, 


respectively,   the   work   done   by  the   substance   will   be 

7  dvp  , 

Aa .  dv,  or  r    7£  dv. 
do 

But  we  have  already  seen,  in  Art.  51,  that  the  efficiency 
of  the  cycle  is  ^.     Hence 


deQ  , 
,--dv 
dv 


~dv 


.(55). 


This  remarkable  result  was  practically  obtained  by  Carnot, 
who  found 

d*Q_rdvp 

dv  d0' 

C  being  an  unknown  function,  now  called  Carnot 's  func- 
tion, which  was  independent  of  the  nature  of  the  working 
substance. 

P.  16 


242  ELEMENTARY    THERMODYNAMICS. 

Substituting  in  equations  (53)  and  (54),  we  get 


M-        i (56)- 


and  therefore 


Expressing  the  condition  either  that  dU  or  that  d<f>  is 
a  complete  differential,  we  obtain  the  curious  result 
deOv        d\p 
~d;=e~dP   ..................  (°7)' 

whatever  the  substance  may  be. 

Again,  if  Cpdd  be  the  heat  required  to  raise  the 
temperature  of  the  body  from  6  to  6  +  dO,  Cp  may  be 
called  the  specific  heat  of  the  body  at  constant  pressure, 
and  the  fundamental  equation  dU=dQ—pdv  becomes, 
when  p  is  constant, 

dU=Cpd6-pdv, 

or  Cpd6  =  dU+pdv 

=  d(U+pv), 


d6 

Hence  if  we  take  d  and  p  as  independent  variables,  we 
have  in  any  reversible  operation, 


=  d(U  +  pv)-vdp 

( U  +  pv\ 

— j v 

dp 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE. 


243 


To  find  ~,  we   take   the   same   reversible   cycle  as 
before,  and  draw  Aof,  Bb'  parallel  to  Ov  to  meet  CD. 

P 


Hence  the  area  of  ABCD  may  be  written 

—  Aaf .  dp, 

for  the  ordinates  of  A  and  B  are  p  and  p  +  dp  respectively. 
Substituting  r  -^  for  Aaf,  this  becomes 

dpv  7 
-Td6dp' 

The  heat  absorbed  during  the  operation  represented  by 
A  B  being  -*—  dp,  we  have  therefore 


dp- 

deQ  _      a  dpv 


-dp 


-6 


(58). 

16—2 


244 
Hence 


ELEMENTARY   THERMODYNAMICS. 


dQ 

JA  -  C  —  -  d- 
6      d 

dU=dQ-pdv 


dp, 


(59); 


and  since  dU  a,ud  d$  are  complete  differentials, 


.(60). 


94.  If,  in  equation  (56),  we  suppose  that  6  and  v  vary 
in  such  a  manner  that  p  remains  constant,  we  shall  have 
dQ  =  CpdO  and  therefore 


But  since  dp=~ 

we  have  0  =  %+^"gT. 

d0       dv  dO 

Hence  Cp-  Cv=-  6  ^4^-....  ...(61). 

fdep\ 

\dv) 

If  6  and  p  be  taken  as  independent  variables,  we  shall 
obtain 


cp-cv=-e 


.(62). 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE. 
For  the  ideal  perfect  gas,  pv  =  R6,  so  that 


245 


_  =  =  _=_ 

d6      v  '    dd      p'     dv         v2  '' 

and  therefore,  by  equations  (57),  (60),  and  (61),  Cp  and  Cv 
are  functions  of  6  only  such  that 

CP-C0  =  R 
Also,  by  equations  (56), 

dU=Gvdd 


Hence  also,  since  dU  is  a  complete  differential,  we  see 
that  both  C,  and  U  can  only  vary  with  0,  just  as  in 
Chap.  II. 

95.  If  the  substance  at  pressure  p,  volume  v,  and 
temperature  6,  be  compressed  by  an  increase  of  pressure 
dp  to  volume  v  +  dv,  the  Isothermal  Compressibility,  Ke, 
is  denned  to  be 

—  dev 

— 
ep  v  dp' 

and  the  Adiabatic  Compressibility,  K+,   ...............  (63). 


K 

~ 


-ft-  A  == 


1  d^v 
v  dp  ' 


The  reciprocal  of  the  compressibility  may  be  called  the 
Elasticity  of  volume,  and  be  denoted  by  E  :  thus 


..(64), 


dv 


246  ELEMENTARY  THERMODYNAMICS. 

d*p 

T  ^       f  E&      dv 

and  therefore  -pr  =  •? — • 

Eg        dgp 

dv 
But  we  have 


so  that,  by  the  thermodynamic  relations 

d*P  _  ^y  ^ 
~dv  ~  dd  dv  ' 

d^ 
dv 

~d^' 
dp 


and  ? 

dv       d<f>  dv 

dvp 


Hence  ^-  dv  '  d« 

£0      dK(f>  dvp 

dp  '  dd 


since  from  the  equation 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE. 

dp<f>     dp(j)   dpv 

weset  di-dZ-W' 

dv$     dv$  dvp 

and  -dJ  =  -dp'Te- 

Now  from  the  equation  dQ  =  0d<}>,  we  find 


Thus  we  have  the  important  relation 


247 


/QK\ 

(65)' 


Again,  we  define  e,  the  '  coefficient  of  cubical  dilatation 
by  heat  at  constant  pressure  ',  to  be  such  that 


But  from  the  relation 


,     .  dvp     dep  dpv 

obtain  0  =  -y£  +  ~r-  -?a  . 

dQ       dv  dQ 


,     . 

<66>- 


v  /n-. 

Hence  e  =  --  -r—=  -FT^£  ................  (67). 

v  dep     EQ  dQ 

dv 
In  terms  of  these  definitions,  we  obtain 

(68). 


248  ELEMENTARY  THERMODYNAMICS. 

For  the  ideal  perfect  gas,  pv  =  R9,  and  therefore 
1 


For  liquids  and  solids  at  constant  pressure,  we  may 
generally  write 

v  =  v'  (1  +  a<9'), 

for  moderately  small  changes  of  temperature,  where 

0'  =  6  -273, 

v'  is  the  value  of  v  at  0°  C.,  and  a  a  small  number  in- 
dependent of  6. 

Consequently,       v  =  v'(l-  273a)  (1  +  a0), 
and  log  v  =  log  [v'  (1  -  273a)]  +  a#,  nearly. 

Hence         e  =  -  -^  =  a,  a  function  of  p  only. 

Solid  bodies,  in  expanding  by  heat  at  constant  pressure, 
preserve  their  forms.  Thus  if  Z  be  the  length  of  the  edge 
of  a  cube  at  temperature  6,  we  may  write 


where  I'  is  the  value  of  I  at  0°  C.,  and 

1  +  «<9'  =  (1  +  J36J, 
or  a  =  3/3. 

The  small  number  ft  is  called  the  'coefficient  of  linear 
expansion.' 

96.     The  value  of  equation  (68)  may  be  illustrated  by 
the  case  of  water  ;  for  Cv  cannot  be  determined  for  water 


APPLICATIONS  OF   CARNOT's   PRINCIPLE.  249 

by  direct  experiment.  Thus  if  J  be  the  number  of  ergs 
in  a  calorie,  or  '  Joule's  equivalent,'  we  have  at  a  pressure 
of  one  atmo  : 


at  25°  C.,...Cp  =  Jx  1-0016,.  ..e  = 

at  50°  C.,...C^  =  Jx  1-0042,...  e  =  -00049,...  v 

Also  when  the  pressure  is  one  atmo,  the  diminution  of 
volume  due  to  an  increase  of  pressure  of  one  atmo  is  found 
to  bear  to  the  original  volume  the  ratio  : 

at    0°C  .......  -00005.  ] 

,  at  25°  C  .......  -000046.1 

at  50°  C  .......  -000044.  j 

Hence  if  n  be  the  number  of  dynes  per  square  centimetre 
in  a  pressure  of  one  atmo,  we  obtain,  for  a  pressure  of  one 
atmo  : 

at    °°C  .......  £« 


at30°C  .......  *»  =  .  000044 

Thus  at  a  pressure  of  one  atmo,  we  have  for  water 
at    0°C  .......  ^-a  = 

at25°C  .......  Cp-C0  = 

at  50°  C  .......  Cp-CD  =  Jx  -0425, 

and  therefore  : 

at    0°C  .......  C^Jx-99956.) 

at  25°  C  .......  0B  =  /x-9941.   [ 

at  50°  C  .......  C0  =  Jx-9617. 


250  ELEMENTARY   THERMODYNAMICS. 

97.  When  the  substance  undergoes  a  reversible 
operation  in  which  heat  is  neither  gained  nor  lost,  the 
equation 


gives  de  =  -edv  .....................  (69). 

If  we  take  the  equation 


we  get  de  =  ~dp  ..................  (70). 

Up 

If  then,  when  the  pressure  is  kept  constant,  the  sub- 
stance contracts  as  its  temperature  rises,  like  water 
between  the  freezing  point  and  its  point  of  maximum 
density,  e  will  be  negative,  and  in  an  isentropic  operation, 
d6  and  dp  will  have  opposite  signs,  so  that  an  increase  of 
pressure  will  cause  a  fall  of  temperature. 

98.  We  will  now  explain  the  experiment  by  which 
Joule  and  Thomson  determined  more  exactly  the  absolute 
temperature  of  the  freezing  point  arid  the  true  law  of 
what  are  called  perfect  gases. 

A  stream  of  gas  is  kept  constantly  flowing  by  means  of 
a  pump  through  a  long  pipe  in  one  short  length  of  which 
there  is  firmly  fixed  a  porous  plug  of  cotton-wool  or  waste 
silk,  by  which  the  motion  of  the  gas  is  so  impeded  that 
its  velocity  remains  small  even  when  there  is  a  consider- 
able difference  between  the  pressures  before  and  behind 
the  plug.  If  the  pump  be  at  a  sufficient  distance  and 
worked  as  steadily  as  possible,  the  pulsations  which  it- 
causes  will  be  imperceptible,  and  after  a  little  time  from 


APPLICATIONS   OF   CARNOT's   PRINCIPLE.  251 

the  commencement  of  the  experiment,  the  state  of  the 
motion  will  become  steady  except  in  the  immediate 
neighbourhood  of  the  plug. 

Suppose  then  that  P  and  Q  are  two  sections,  one 
before  and  the  other  behind  the  plug,  but  at  such  a  dis- 
tance from  it  that  the  irregular  motions  and  pressures  due 


Plug. 

to  the  passage  through  it  are  not  discernible,  so  that  there 
is  only  a  uniform  current  of  gas  to  be  considered.  And 
let  the  portion  PQ  of  the  pipe  be  surrounded  by  some 
non-conducting  material,  so  that  heat  can  neither  enter 
nor  escape  through  it. 

Let  p  and  6  be  the  constant  pressure  and  temperature 
and  v  the  constant  volume  per  gramme,  of  the  gas  which 
passes  through  the  section  P,  and  let  (p',  v',  &}  be  the 
values  of  the  same  quantities  at  the  section  Q.  Then 
since  we  may  imagine  that  the  entering  stream  is  forced 
towards  the  plug  by  an  ideal  piston  at  P,  and  that  the 
emergent  stream  forces  out  another  ideal  piston  at  Q,  it 
is  clear  that  when  one  gramme  of  gas  passes  through  the 
portion  PQ  of  the  pipe,  the  external  work  done  upon 
it  will  be 

pv  —  p'v'. 

But,  since  the  energy  of  the  mechanical  motion  is  practi- 
cally zero,  the  change  of  energy  is  given  by  the  equation 


252  ELEMENTARY   THERMODYNAMICS. 

Hence  the  heat  absorbed  by  the  gas  is 


and  as  in  the  experiment  this  is  zero,  we  have 


Now  the  expression  on  the  right  hand  side  is  merely  the 
quantity  of  heat  in  ergs  that  is  required  to  raise  the  tem- 
perature of  one  gramme  of  the  gas  at  constant  pressure 
from  6  to  6',  and  its  determination  does  not  require  us  to 
possess  any  previous  knowledge  of  absolute  temperature. 
In  order  to  find  it,  the  difference  of  temperature  between 
the  sections  P  and  Q  was  measured  by  Joule  and  Thomson 
by  a  mercury  thermometer,  but  the  very  same  thermo- 
meter had  been  previously  used  by  Joule  in  determining 
the  value  of  J,  that  is,  the  number  of  ergs  in  a  calorie. 
The  value  of  Cp,  or  the  specific  heat  in  calories  multiplied 
by  J,  was  deduced  from  Regnault's  experiments.  The 

rQ 
product  was  the  integral       Cpd0,  since  &  —  6  was  small. 

J  P 

The  experiments  showed  that  for  a  given  value  of  0,  the 
integral  was  simply  proportional  to  p  —  p,  not  merely  for 
an  infinitesimal  difference  of  pressure,  but  for  differences 
of  5  or  6  atmos. 

In  the  case  of  hydrogen,  the  gas  was  slightly  heated 
by  passing  through  the  plug  and  the  heating  effect  was 
observed  at  temperatures  from  4°  C.  or  5°  C.  to  about 
90°  C.  The  investigation,  however,  was  not  carried  out 
in  sufficient  detail  to  give  any  law  of  variation  of  this 
small  effect  with  temperature,  and  we  shall  therefore 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE.  253 

take  the  mean  of  the  results,  which  may  be  expressed 
thus: 


f 

Jf 


p 

where  cp  is  the  specific  heat  in  calories,  and  a   the  con- 
stant number  '000  000  13116.     In  other  words, 


where  a  =  '000  000  13116  x  J  =  5'448. 
Thus  jQp(ddjjj-v)dp=a(p-p'), 

and  therefore 

0^ -v=-a  (71). 

dp  (v\         a 

Hence  JU)=-£" 

or  v  =  a  +  0f(p) (72), 

where /(p)  is  an  unknown  function  of  p. 

From  this  equation  we  have 

pv  =  ap  +  Qpf(p). 

Xow  we  know  that  when  the  temperature  is  high  enough 
and  the  pressure  not  too  great,  every  perfect  gas  satisfies 
very  approximately  a  relation  of  the  form  pv  =  Rd.  The 
quantity  a  cannot  therefore  be  strictly  constant,  as  we 
have  supposed.  But  it  appears  from  Joule  and  Thomson's 
experimental  results  that  it  varies  little  between  the 
temperatures  4°  C.  and  90°  C. 

If  the  absolute  temperature  of  the  freezing  point  be 
denoted  by  00»  the  absolute  temperature  of  the  boiling 
point  will  be  80  + 100.  Hence,  if  v0  and  vm  be  the  cor- 


254  ELEMENTARY  THERMODYNAMICS. 

responding  values  of  v  at  any  the  same  pressure  p,  we 
shall  have 


and  vm  -  v»  =  100/(p). 

mi.      f  v0—a00 

Therefore  -  =  7-^  , 

^100  —  ^0       100 

j.1       -     •  $0  ^0  /i  « 

that  is,  •-£-  =  —      -     1  -  - 

100  V1W-V0\  VQ 


where  E1  stands  for 


Now  for  hydrogen  expanding  at  the  constant  pressure  of 
one    atmo,    Regnault    found    ^  =  '36613,   and   therefore 

100 

-vr  =  273*13.       Also,    at    the    pressure    of    one    atmo, 

v0  =  11164-45.     Thus  we  get 


=  27313  (1  -  -000488) 

=  273. 

In  the  case  of  common  air  and  carbonic  acid,  the 
thermal  effect  observed  was  a  slight  lowering  of  tempera- 
ture, which  was  shown,  in  1862,  to  vary  at  different  tem- 
peratures very  nearly  in  the  inverse  ratio  of  the  square  of 
C  +  273,  where  G  is  the  temperature  at  either  the  section 
P  or  the  section  Q,  as  shown  by  the  mercury  thermometer. 
Thus  we  have 


E  must  not  be  confused  with  E.  or  E 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE.  255 

that  is, 


.  /273V  ,          ,x 

or  =-b(-gr)    (p-p)- 

From  either  of  these,  remembering  that  the  variation  of  6 
with  p  is  small,  we  get 

0i?_w=6(273y 


V 

/273N2 


(IH^j- 


and  therefore 


Hence  B»-  +^/(p)  ...............  (74), 

and  as  before 


so  that,  if^s-'l0 


^0_      6 
TOO-  +^ 

In  the  small  terms  on  the  right  hand  side  of  this  equation 
we  may  put  00  =  273.     We  have  then  the  formula 

100  f.       6  /.    ,  -464V)  ^.s 

^-1  +       1+          .........  (7o)' 


256  ELEMENTARY  THERMODYNAMICS. 

Now  for  common  air,  b  =  2'684  ;  and  at  the  pressure 
of  one  atmo,  E  =  "36706,  and  v0  =  773'3.  The  correspond- 
ing quantities  for  carbonic  acid  are  b  =  12'323,  E=  '37100, 
and  v0  =  505-7.  Thus  we  obtain  in  the  two  cases, 

00  =  272-44(1  +  -00261), 
and  00  =  269-5(1+  -0182), 

that  is,  00  =  272-44  +  71  =  273-15, 

and  00  =  269-5  +  4'90  =  274'4, 

respectively. 

The  results  obtained  from  the  three  gases  are  collected 
in  the  accompanying  table. 


Uncorrected  estimate  of 

absolute  temperature       Correction.        Result, 
of  the  freezing  point. 


Hydrogen  -36613  273°'13  --13°        273«'00 

Air  i  -36706  272-44  +-71          273'15 

Carbonic  acid      -37100  '  269'5  +4-90     .  274-40. 

99.     Again,  from  equation  (74),  we  have 

(O'TQx  2 
I r)  - 

But  since,  when  the  temperature  is  high  enough,  the  gas 
satisfies  the  relation  pv  =  R8,  it  is  evident  that  pf(p)  =  E, 
a  constant.  Thus,  finally, 

pv  =  Re-ibp(2~J    (76). 

This  equation,  according  to  Thomson  and  Joule,  must  be 
used  instead  of  the  simpler  relation  pv  =  R6. 


APPLICATIONS   OF  CARNOl's   PRINCIPLE.  257 

An   exactly  similar  equation  had   been   obtained  by 
Rankine  in  1854,  in  the  form 

*>•-•»»- 


where  a  is  a  constant. 

D/D 

By  substituting  --   for  p  in  the  small  term  of  equation 

(76),  the  two  will  easily  be  seen  to  be  identical. 
From  equation  (76),  we  find 

rfpV  =  JR  +  1&  /273^2 

and  -  *? : 

b  /273\2 


Also  since  pv\l  +  £-  (  n  V   =  Rd, 

(       3v  \  u  J  } 


,-eobtam 


Hence 

...(79). 


deCp_  _  Q  d-pv  _  26  /273V 

As   an   illustration,   let   cp  be   the    specific    heat    at 
constant  pressures  in  calorics,  P  the  pressure  in  atmos, 
J  the  number  of  ergs  in  a  calorie,  and  n  the   number 
P.  17 


258  ELEMENTARY  THERMODYNAMICS. 

of  dynes   per   square   centimetre   in   a   pressure   of  one 
atmo.     Then 

dp~ J  dp    je  (  e  ) ' 

Thus  for  common  air, 

decp_'1308    /273V 

~dP~    e   '  (  e  )  ' 

and  for  carbonic  acid, 

dgCp_-Q    /273V 

dP     d'\e  )  ' 

I  dvv     R      26  /273V 


and 


100.  As  a  further  illustration  of  Carnot's  principle, 
we  will  consider  the  case  of  a  strained  rod,  the  straining 
force  being  so  great  that  the  pressure  of  the  atmosphere 
and  the  weight  of  the  rod  may  be  neglected. 

Let  T  be  the  force,  considered  positive  when  tensional, 
I  the  length  of  the  rod  when  acted  on  by  the  force  and  0 


APPLICATIONS   OF   CARNOT'S    PRINCIPLE. 


259 


its  uniform  temperature.     Then  in  any  reversible  process, 
taking  6  and  T  as  independent  variables,  we  have 


where  M  is  the  mass  of  the  rod  and  CT  its  specific  heat 
under  constant  tension. 

To  find    -^ ,  let  the   rod  be  made  to  undergo  the 
ct  J. 

following  complete  cycle  of  reversible  operations. 

(1)  Let    the    rod   be   slowly   stretched   at   constant 
temperature  until  T  increases  by  dT.     The  heat  absorbed 

win  be  ^r. 

(2)  Let  the  rod  be  still  further  stretched,  but  without 
loss  or  gain  of  heat,  until  the  temperature  becomes  0  —  r, 
where  T  is  indefinitely  small. 


(3)     Then,  whilst  the  temperature  is  kept  constantly 
equal  to  0—  r,  let  the  tensions  be  slowly  reduced  by  such 

17—2 


260  ELEMENTARY  THERMODYNAMICS. 

an  amount  that  an  adiabatic  operation  will  restore  the  rod 
to  its  original  state. 

If  the  cycle  be  represented  by  a  diagram  in  which  T 
and  I  are  independent  variables,  it  will  easily  be  seen  that 
the  work  done  by  the  rod  during  the  cycle,  is 


We  have  therefore 


Thus  dQ  =  MCTd0  +  0^~dT (84), 

and  therefore,  since  dQ  =  0d<j>, 

The  condition  that  d<f>  is  a  complete  differential,  is 

o^CMQ       dy 

dT   ~        dP  (  b)' 

In    any   adiabatic   operation.   dQ   and   d<f>   are    both 
zero:  hence 

%--*,3 <*> 

This  relation  was  first  obtained  by  Sir  W.  Thomson  and 
led  him  to  make  a  curious  prediction  with  respect  to  the 
behaviour  of  india-rubber,  which  was  experimentally 
verified  by  Joule  in  1859. 


APPLICATIONS   OF   CARNOT'S   PKINCIPLE.  261 

So  long  as  india-rubber  is  acted  on  by  no  force  or  only 
by  a  small  force,  it  exhibits  the  same  phenomena  as  most 
other  substances,  lengthening  when  heated  and  shortening 
when  cooled.  But  when  the  force  is  great  enough,  it 
shortens  when  heated  and  lengthens  when  cooled.  Under 

these  circumstances,   jt  is  negative,  and  therefore,  since 

J      /\ 

CT  is  positive,  we  see  that  -vU  must  be  positive.     India- 
rubber  will  therefore  be  heated  by  an  increase  of  the  strain- 
ing force  when  that  force  is  large  enough,  and  conversely. 
Again,  equation  (84)  may  be  written 


Hence,  since  Q  =  ^  +  dTde (88)' 

we  have 


Thus  if  we  denote  by  Ct  the  specific  heat  of  the  rod  at 
constant  length,  we  obtain 

d'Tdi 

•(89); 


and  C,-Cr  +  (90), 


262  ELEMENTARY   THERMODYNAMICS. 

or,  by  equation  (88), 

0 


dl 

W 

0\d0) 
I=*-M—' 

dT 

AT  d&d       0    d{T  /m\ 

A1S°  -a-m,dS  ..................  (91)' 

and  the  condition  that  d<f>  is  a  complete  differential,  gives 


Lastly,  taking  T  and  I  as  independent  variables,  we 
have  from  equation  (89), 


or,  by  equation  (90), 

TT  wrf>-^  \J  rr       dl 

Hence  — 2_  =  _  .JL  __ 

dl  d  drf  ' 

dT 

or  _*-    =  — I    g  (()d,\ 

dl       d    dl  " 

Substituting  for  Gt,  we  obtain 

'<yy 


APPLICATIONS   OF  CARNOT's   PRINCIPLE.  263 

PART   II. 

CHANGE   OF  AGGREGATION. 

101.  Bodies  are  found  in  three  different  states  of 
aggregation,  known  as  the  solid,  the  liquid,  and  the 
gaseous.  Most  substances  are  capable  of  existing  in  all 
three  states.  For  example,  water  exists  in  the  forms  of 
ice,  water,  and  steam.  A  few  solids  have  not  yet  been 
melted,  but  the  number  of  such  bodies  is  found  to  diminish 
as  improvements  in  the  Arts  and  Sciences  place  higher 
temperatures  at  our  disposal.  Prior  to  1877,  the  more 
perfect  gases  oxygen,  hydrogen,  and  nitrogen,  had  resisted 
all  attempts  to  reduce  them  to  the  liquid  state;  but  in 
that  year  they  were  not  only  liquefied  but  solidified,  by 
two  independent  experimenters,  M.  Cailletet  and  M.  Raoul 
Pictet.  It  is  therefore  concluded  that  when  we  shall  have 
a  sufficient  range  of  temperature  and  pressure  at  our 
command,  it  will  be  possible  to  make  every  substance 
take  the  three  different  forms  of  solid,  liquid,  and  gaseous. 

Let  any  quantity  of  any  substance,  as  water,  be  con- 
tained in  a  cylinder  fitted  with  an  air-tight  piston,  so  that 
the  volume  can  be  increased  or  diminished  at  pleasure  ; 
and  suppose  that,  owing  to  the  moderate  size  of  the 
cylinder,  the  weight  of  its  contents  produces  no  sensible 
difference  of  pressure  in  any  part  of  it.  Then  it  is  found 
that  if  the  substance  at  any  given  temperature  can  exist 
in  stable  equilibrium  in  two  different  states  in  any  propor- 
tion, it  is  possible,  by  altering  the  volume,  to  cause  it  to 
exist  in  stable  equilibrium  in  these  two  states  at  the  same 


264  ELEMENTARY  THERMODYNAMICS. 

temperature  and  pressure  in  all  other  proportions.  We 
cannot  therefore  take  6  and  p  for  independent  variables 
to  define  the  state  of  the  system  when  in  equilibrium. 

102.  When  a  solid  body  is  raised  to  a  sufficiently  high 
temperature,  it  begins  to  melt  into  a  liquid.  This  change 
of  state  often  takes  place  abruptly,  as  when  ice  is  con- 
verted into  water;  but  sometimes  there  are  indications 
of  an  approaching  change  of  state  before  that  change 
actually  occurs.  For  instance,  glass,  before  reaching  a 
state  of  perfect  liquefaction,  passes  through  a  series  of 
intermediate  states  in  which  it  is  soft,  or  viscous,  and  can 
readily  be  drawn  out  into  very  fine  threads.  But  in  all 
cases,  when  a  body  exists  in  the  same  vessel  in  a  state  of 
stable  equilibrium  at  a  given  temperature  and  pressure, 
partly  in  the  solid  and  partly  in  the  liquid  state,  the 
proportion  of  the  two  parts  may  have  any  value  we  please. 
This  temperature  is  called  the  melting  point  of  the  sub- 
stance for  the  given  pressure. 

By  carefully  cooling  a  liquid  in  a  clean  vessel,  it  is 
found  possible  to  reduce  the  temperature  below  the  melt- 
ing point  without  causing  it  to  solidify.  The  liquid  is 
then  in  a  state  of  unstable  equilibrium;  the  smallest  shake 
or  touch  causing  it  to  solidify  with  explosive  violence. 

Many  solid  bodies  are  constantly  in  a  state  of  evapora- 
tion, or  of  transformation  into  the  gaseous  state.  Camphor 
and  ice  are  the  best  known  examples  of  this.  Such  bodies, 
if  not  kept  in  well-stoppered  bottles,  gradually  escape  in 
the  form  of  vapour.  For  example,  large  sheets  of  ice 
during  a  long  dry  frost  become  smaller  and  at  last  dis- 
appear. Very  little  is  known  of  the  conditions  attending 
this  phenomenon  except  from  theoretical  considerations. 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE.  265 

Again,  if  heat  be  applied  to  a  liquid  contained  in  a 
closed  vessel,  part  of  it  "will  be  converted  into  vapour  or 
gas,  and  it  is  found  that  the  pressure  of  the  vapour  when 
in  stable  equilibrium  with  the  liquid,  depends  only  on  the 
temperature,  increasing  as  the  temperature  rises.  If  the 
space  above  the  liquid,  instead  of  containing  nothing  but 
the  vapour  of  the  liquid,  contain  any  quantity  of  air  or 
any  other  gas  not  capable  of  chemical  action  on  the  liquid, 
it  was  found  by  Dalton  that  the  quantity  of  vapour  formed 
was  very  nearly  as  great  as  in  the  first  case,  but  that  the 
time  required  to  reach  the  state  of  equilibrium  was  much 
longer. 

103.  As  the  transformation  from  the  liquid  to  the 
gaseous  state  has  been  most  studied  experimentally,  and 
as  the  phenomena  are  the  same  in  all  cases,  we  will 
describe  the  boiling  of  water  by  way  of  example. 

When  water  is  heated  in  an  open  vessel,  the  lowest 
layer  gets  hot  first,  and  by  its  expansion  becomes  lighter 
than  the  colder  water  above  and  gradually  rises,  so  that  a 
gentle  circulation  is  kept  up  whereby  the  whole  of  the 
water  is  warmed,  though  the  lowest  layer  is  always  the 
hottest.  As  the  temperature  increases,  the  air,  which  is 
always  absorbed  to  a  small  extent  by  cold  water,  is  silently 
expelled,  rising  to  the  surface  in  small  bubbles.  At  length, 
the  lowest  layer  of  water  becomes  so  hot  that  in  spite  of 
the  pressure  of  the  atmosphere  and  the  weight  of  the 
water  above,  a  bubble  of  steam  is  formed  which  rapidly 
grows  larger,  and  then  rises,  but  is  condensed  by  the 
colder  water  into  which  it  ascends,  the  collapse  producing 
the  well-known  noise  of  'simmering'  or  'singing.'  In 
this  way,  the  water  is  briskly  agitated  until  it  becomes  hot 


266  ELEMENTARY  THERMODYNAMICS. 

enough  throughout,  and  then  the  bubbles  rise  to  the 
surface  and  the  water  is  said  to  boil. 

The  temperature  at  which  pure  water  boils  depends 
chiefly  on  the  pressure  of  the  atmosphere,  the  greater  the 
pressure  the  higher  being  the  boiling  temperature.  But 
the  water  must  be  raised  to  a  higher  temperature  than 
that  at  which  the  pressure  of  the  steam  is  equal  to  that 
of  the  atmosphere,  for  the  bubbles  of  steam  have  to  over- 
come not  only  the  pressure  of  the  atmosphere,  but  the 
weight  of  the  water  itself.  The  dependence  of  the  boiling 
point  on  the  pressure  may  be  clearly  shown  by  connecting 
a  closed  glass  vessel  containing  water  and  air  with  an  air- 
pump  by  means  of  which  the  pressure  of  the  air  can  be 
reduced  at  pleasure.  After  working  the  pump  a  short 
time,  the  water  will  be  seen  to  enter  into  active  ebullition. 
In  this  way  it  may  be  made  to  boil  at  any  temperature 
between  0°  C.  and  100n  C.  On  the  other  hand,  by  increas- 
ing the  pressure  of  the  air,  the  boiling  point  may  be  raised 
above  100°  C. 

A  simpler  experiment  is  due  to  Franklin.  A  little 
water  is  boiled  in  a  flask  over  a  gas  flame  until  most  of 
the  air  dissolved  in  it  is  expelled ;  and  while  the  water  is 
still  boiling,  the  cask  is  securely  corked  and  removed  from 
the  flame.  Ebullition  then  ceases,  but  if  cold  water  be 
poured  over  the  vessel  so  as  to  condense  the  steam,  the 
water  again  begins  to  boil  and  continues  to  do  so  for  a 
considerable  time. 

104.  In  order  to  explain  the  indicator  diagram  of  a 
substance,  part  of  which  is  liquid  and  part  vapour,  we  will 
suppose  a  gramme  of  water  contained  in  a  cylinder  fitted 
with  an  air-tight  piston.  If  the  interior  of  the  cylinder 


APPLICATIONS   OF   CARNOT  8   PRINCIPLE. 


267 


be  large  enough,  the  whole  of  the  water  will  exist  as  steam 
satisfying  approximately  the  relation  pv  =  Rd,  where  R  is 
the  constant  4,752,300.  If,  while  the  temperature  is  kept 
constant,  we  then  force  in  the  piston  so  as  to  diminish  the 
capacity  of  the  cylinder,  the  product  pv  will,  at  length, 
begin  to  decrease,  and  continue  to  decrease  until  the 
density  of  the  vapour  is  exactly  equal  to  that  which  is  in 
stable  equilibrium  with  water  at  the  same  temperature. 
The  steam  is  then  said  to  be  saturated  and  the  smallest 
increase  of  pressure  is  sufficient  to  cause  some  of  it  to  be 
condensed  into  water.  We  may,  if  we  are  careful,  convert 
all  the  steam  into  water  without  any  appreciable  change 
either  in  the  temperature  or  the  pressure. 

The  state  of  the  substance  before  condensation  begins 
is  represented  on  the  indicator  diagram  by  the  curve  AB, 


and  during  the  change  of  state  by  the  horizontal  line  BC. 
Also  since  water  is  very  nearly  incompressible,  the  re- 
mainder of  the  diagram  will  be  the  practically  vertical 
line  CD. 

When  the  steam  is  in  the  state  represented  by  the 


268  ELEMENTARY   THERMODYNAMICS. 

point  B,  it  is  possible  to  keep  the  temperature  constant 
and  yet  to  increase  the  pressure  without  causing  any 
steam  to  condense  until  the  state  represented  by  the 
point  X  is  reached.  In  like  manner,  the  pressure  may  be 
reduced  considerably  below  that  indicated  by  the  point  C 
before  any  water  evaporates.  But  in  both  cases,  the 
substance  will  be  in  an  unstable  condition,  being  in  danger 
of  explosive  condensation  on  the  curve  BX,  and  of  explo- 
sive evaporation  on  the  curve  CZ. 

It  has  been  suggested  by  Prof.  J.  Thomson  that  the 
curves  BX  and  CZ  may  be  continued  into  one  another,  as 
by  the  dotted  line  XYZ,  so  that  the  isothermal  is  only 
apparently  and  not  really  discontinuous.  But  experi- 
mental evidence  of  the  existence  of  the  curve  XYZ  is 
still  wanting. 

It  will  be  observed  that  at  any  point  in  the  curve 
XYZ,  the  pressure  and  the  volume  increase  or  diminish 
together,  so  that  the  state  of  the  substance  is  then  essen- 
tially unstable.  In  the  present  chapter,  we  restrict  our- 
selves to  stable  conditions  and  reversible  operations,  and 
therefore  the  discontinuous  part  of  the  diagram  is  all  we 
require. 

In  the  accompanying  figure,  the  isothermals  for  diffe- 
rent temperatures  are  collected  together.  The  dotted 
curve  B  B'  B"...  indicates  the  pressures  and  the  volumes 
of  saturated  steam  and  is  therefore  called  the  '  steam  line.' 
It  is  not  an  isothermal ;  for  if  it  were,  the  different  iso- 
thermal curves  AB,  A'B',  A"B",...  would  all  coincide  with 
it,  and  for  any  given  pressure  and  volume,  the  temperature 
of  one  gramme  of  steam  might  have  an  infinite  number  of 
values,  which  is  plainly  absurd.  It  is  worthy  of  notice 
however,  that  a  finite  number  of  different  temperatures 


APPLICATIONS   OF   CARNOT's   PRINCIPLE. 


269 


are  sometimes  possible  when  the  pressure  and  volume  of 
a  given  mass  of  a  substance  are  given.     For  example,  at  a 
P 


pressure  of  one  atmo,  a  given  quantity  of  water  will  have 
the  same  volume  when  its  temperature  is  a  little  below 
4°  C.  as  when  it  is  a  little  above. 

The  liquid  state  is  practically  represented  by  the  axis 
of  p,  the  volume  of  a  given  quantity  of  water  at  any 
ordinary  temperature  being  negligible  in  comparison  with 
the  volume  of  the  same  quantity  of  saturated  steam.  The 
axis  of  p  is  therefore  the  '  water  line.' 

Along  the  steam  line,  there  is  evidently  a  relation 
between  p  and  0  which  may  be  written  p=f(0}.  A 
similar  relation  will  hereafter  be  found  to  hold  between 
p  and  6  when  the  solid  and  the  liquid,  or  the  solid  and  the 
gaseous  states  are  in  stable  equilibrium  together  in  the 
same  vessel. 

The  steam  line  for  water  is  represented  approximately  by 
either  of  the  following  empirical  formulae,  due  to  Rankine  : 


constant.  | 

v  =  (6  -  233)5  x  constant,  j  ' 


270  ELEMENTARY   THERMODYNAMICS. 

Since  the  density  of  saturated  steam  increases  with 
the  temperature,  it  follows  that  the  steam  and  water  lines 
continually  approach  one  another,  and  we  are  naturally 
led  to  ask  if  they  ever  meet.  This  question  has  been 
answered  by  the  experiments  of  Cagniard  de  la  Tour  and 
Dr  Andrews,  from  which  it  appears  that  for  each  substance 
there  is  a  certain  temperature,  known  as  its  'critical 
point,'  above  which  the  distinctions  between  the  liquid 
and  gaseous  states  disappear,  whatever  the  pressure  may 
be. 

105.  Let  any  sufficient  quantity  of  any  liquid  and  its 
saturated  vapour  be  contained  in  a  state  of  equilibrium 
in  a  cylinder  fitted  with  an  air-tight  piston ;  and  suppose 
that  the  cylinder  is  not  so  great  but  that  the  effect  of 
gravity  in  creating  differences  of  pressure  within  it  may 
be  neglected.  Then  if,  by  slowly  drawing  out  the  piston, 
an  additional  quantity  of  saturated  vapour  is  formed  in  a 
reversible  manner  without  altering  either  the  temperature 
or  the  pressure,  it  is  evident  that  the  heat  absorbed  will 
be  proportional  to  the  quantity  of  saturated  vapour  thus 
produced,  whether  we  adopt  the  caloric  or  the  true  theory 
of  heat.  The  heat  absorbed  when  an  additional  gramme 
of  saturated  vapour  is  formed  in  a  reversible  manner  at 
any  constant  temperature  6  and  constant  pressure  p  is 
called  the  Latent  Heat  of  the  vapour  and  is  usually 
denoted  by  the  symbol  L.  Since  there  is  a  relation 
between  p  and  6  along  the  steam  line,  we  may  consider 
L  to  be  a  function  of  6  alone,  or  of  p  alone. 

In  order  to  explain  how  the  subject  was  treated  before 
the  true  theory  of  heat  was  established,  we  will  first 
suppose  heat  to  be  a  material  substance,  so  that  the  total 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE.  271 

quantity  of  heat  absorbed  in  any  cyclical  process  is  zero. 
Suppose  then  that  the  contents  of  the  cylinder  undergo 
the  following  cycle  of  operations. 

(1)  Let   the   piston   be   slowly  drawn  out  until  an. 
additional  gramme  of  saturated  vapour  is  formed  at  con- 
stant temperature  and  pressure,  whereby  a  quantity  of 
heat  L,  is  absorbed. 

(2)  Let  the  piston  be  drawn  further  out  until  the 
temperature  falls  from  0  to  0  —  T,  where  r  is  indefinitely 
small,  just  sufficient  heat  being  imparted  to  the  cylinder 
or  abstracted  from  it,  to  prevent  the  liquid  from  evapo- 
rating and  the  vapour  from  condensing. 

As  we  wish  to  express  the  quantity  of  heat  absorbed 
during  this  operation  in  a  convenient  form,  we  will 
represent  by  C'  the  heat  that  must  be  imparted  to  a 
gramme  of  saturated  vapour  to  keep  it  constantly  in  the 
saturated  state  when  it  is  slowly  compressed  until  its 
temperature  rises  one  degree :  we  will  also  suppose  that, 
under  similar  conditions,  C  denotes  the  specific  heat  of 
the  liquid,  and  H  the  thermal  capacity  of  the  contents  of 
the  cylinder  in  their  original  state.  The  heat  absorbed  in 
the  operation  may  then  be  written 

-(H-C+C')r. 

The  quantity  C'  is  called  the  '  specific  heat  of  the  satu- 
rated vapour';  and  since  most  liquids  are  nearly  incom- 
pressible, it  is  evident  that  C  is  practically  equal  to  Cp . 

(3)  Now  let  the  piston  be  slowly  pushed  in  until  a 
gramme  of  vapour  is  condensed  without  altering  the  tem- 
perature or  the  pressure.     The  heat  absorbed  will  be 


272  ELEMENTARY   THERMODYNAMICS. 

(4)     Lastly,  let  the  contents  of  the  cylinder  be  brought 
into  their  original  state.     The  heat  will  be  Hr. 
We  have  therefore 


(96). 


Naturally,  the  first  liquid  experimented  upon  was 
water.  It  was  ascertained  by  James  Watt  that  the  latent 
heat  of  steam  diminished  as  the  temperature  increased, 
and  he  supposed  his  experiments  to  prove  that  the 
quantity  of  heat  required  to  raise  unit  mass  of  water  from 
the  freezing  point  to  any  temperature  6  at  constant 
pressure  and  then  to  convert  it  into  steam  at  that  tempe- 
rature and  pressure,  was  independent  of  6.  This  con- 
clusion was  known  as  Watt's  law  and  was  expressed  by 
saying  that,  '  The  sum  of  the  free  and  latent  heat  is 
always  constant  '.  In  analytical  language,  this  becomes 

re 
L  +  I     CpdQ  =  constant. 

J273 

Differentiating,  we  obtain 


and  therefore,  by  equation  (96), 

In  accordance  with  this  result,  which  was  long  taken 
to  be  correct,  it  was  thought  that  when  saturated  steam 
expanded  or  was  compressed  in  a  vessel  impermeable  to 
heat,  it  would  continue  at  the  point  of  condensation.  In 
other  words,  the  steam  line  was  considered  to  be  an 
adiabatic  curve. 

In  1847,  Regnault  published  his  experiments  on  latent 


APPLICATIONS   OF   CARNOT's    PRINCIPLE.  273 

heat  from  which  it  appeared  that  Watt's  law  was  not 

re 
strictly  correct,  the  value  of  L  +  I      Cpd8  increasing  with 

J273 

the  temperature.  Regnault's  result,  which  was  expressed 
in  calories,  may  be  stated  in  the  c.  G.  S.  absolute  system  of 
units  thus  : 

L  +  t  °   Cpd6  =  /(606-5  +  -305^), 

J  273 

where  J  is  the  number  of  ergs  in  a  calorie,  and  &  =  0  —  273. 
Differentiating,  we  find 


and  then,  by  equation  (96), 

C'  =  -305  J. 

It  was  therefore  concluded  that  when  saturated  steam 
was  compressed  and  consequently  heated,  it  was  necessary 
to  supply  heat  to  it  from  without  to  keep  it  in  the  state 
of  saturation,  and  conversely,  that  it  would  have  to  part 
with  heat  when  it  expanded.  From  this  it  followed  that 
when  saturated  steam  was  compressed  in  a  vessel  im- 
permeable to  heat,  part  of  it  would  be  condensed  into 
water,  but  that  if  it  was  allowed  to  expand,  it  would 
be  removed  further  and  further  from  the  saturated  state  ; 
in  other  words,  would  become  superheated. 

106.  The  principles  of  the  mechanical  theory  will 
now  be  applied  to  the  reversible  cycle  described  in  the 
last  article. 

On  an  indicator  diagram,  the  first  and  third  operations 
will  be  represented  by  horizontal  straight  lines  AB,  CD, 
and  the  work  done  by  the  substance  during  the  cycle,  by 
the  product  of  AB  and  PQ. 

P.  18 


274  ELEMENTARY   THERMODYNAMICS. 

Now  the  ordinate  OP  represents  the  pressure  p  and 
OQ  represents  p  —  r^,  where  the  value  of  -^  is  to  be 
found  by  experiment  from  the  properties  of  the  steam 


line.  Also  if  s  be  the  volume  of  one  gramme  of  saturated 
steam,  and  a  that  of  one  gramme  of  water,  we  shall  have 
AB  —  s  —  a:  Thus  if  W  be  the  work  done  by  the  substance 
during  the  cycle, 


dp 


Hence  the  principle  of  energy  gives 


.(97). 


Again,  the  increase  of  entropy  in  the  first  operation  is 
and  in  the  third,  -\~-T~(^}\.     In  the  second 


APPLICATIONS  OF   CARNOT'S  PRINCIPLE.  275 

and  fourth  operations  combined,  it  is  (C  —  Cf)  -~  .     We 
have  therefore 


that  is, 

(98). 

„.      dL      „      L    I 
or 

Combining  equations  (97)  and  (98),  we  get  the  im- 
portant result 


Hence  we  find  ~  +  Cp  =  -305,7, 


107.     Now  in  the  case  of  water  Regnault  obtained  the 
experimental  results  : 

L  +  l°    Cpdd  =  J  (606-5  +  -30507), 

'273 

Cp  =  J(\  +  -000  04(9'  +  -000  000  9  0'2). 
dL 

dd 

and 

£  =  /(  606-5  --6950'-  -000  0  2  ff*-  '000  000  3  0/s), 
so  that,  if  we  neglect  the  difference  between  C  and  Cp,  and 
denote  by  c'  the  equivalent  of  C'  in  calories,  we  have 
'  _  w~  -  606-5  -  -695^  -  -OOP  020'2  -  -OOP  OOP  3  ff* 

~273  +  ff 

By  means  of  this  formula,  the  values  of  c'  are  easily 
calculated  as  in  the  table. 

e'     I          0          I         20         I          50  100        I        150        I        200 


c'        -1-916    !    -1-717    !    -1-465    1    -1-133    |    --879    |    -'676 

18—2 


276  ELEMENTARY   THERMODYNAMICS. 

Clausius  finds  that  the  simpler  formula 
£  =  ,7(607-  -7080') 

may  be  substituted  without  serious  error  for  the  more 
complicated  relation  deduced  by  Regnault  from  his 
experiments.  We  then  should  have 

607  -  -7080' 
C=>305~-273Tr-' 

from  which  the  values  of  c'  may  be  calculated  as  before. 

108.     To  find  the  difference  between  C  and  Cp,  we 
take  the  equation 


and  suppose  it  to  refer  to  one  gramme  of  water  in  the 
liquid  state,  the  relation  between  p  and  0  being  the  same 
as  on  the  steam  line.  Thus  we  have 


Taking    the    temperature    100    C.    for   the    purpose    of 
numerical  illustration,  we  then  have,  by  experiment, 

e  =  -000  8, 

or.  since  the  volume  of  one  gramme  of  water  at  100°  C.  is 
practically  one  cubic  centimetre, 

=  •0008. 


Also         .  =  36,300. 


APPLICATIONS  OF  CARNOT's   PRINCIPLE.  277 

Thus  at  100°  C., 

CP-C=  373  x  -000  8  x  36,300. 
But  from  Regnault's  formula,  we  obtain  at  100°  C. 

'Cp=  1-013  J. 
Hence  C  =J  (1  -013  -  "000  2  6). 

It  therefore  appears  that  C  and  Cp  are  so  nearly  equal 
that  no  important  error  can  have  been  introduced  by 
taking  them  to  be  identical. 

109.     For  Carbon  Bisulphide  (CS2),  Regnault  finds 
L  +  f    Cyd0  =  J(90  +  -146  010'  -  -000  412  3  0'2), 

J  273 

f    Cpd0  =  /(-235  230'  +  -000  08150'2) ; 

J  273 

whence  we  obtain 

L=J(W- -089  220' - -000  49380' 2), 
and 
=  -1460!  -  -000  82460'  -  90 -0^- - -QOO  4938^ 

We  then  easily  calculate  the  values  of  c  as  below  : 

6'  0  50  100  150 


c'     '     -'1837     !     - -16001     I     --1406     |     - -1325 
For  Chloroform  (CH  Cls),  Regnault  finds 

L  +  f    Cpd0  =  J(67  +  -13750'), 

•/273 

Cpd0  =  J(-232  350'  + -000  050  720'2); 

273 

and  therefore 

L  =  J~(67  -  -094850'  -  -000  050  720') 

67  -  -094850'  -  -000  050  720' 
and          c='137o--  ff  -, 


278  ELEMENTARY  THERMODYNAMICS. 

whence  the  following  values  of  c'  are  deduced  : 
o  so  100 


+  •0155 


-•1079         --0549     i     -'0153 
For  Ether  (C4H100), 

L  +  f    Cpd0  =  J(94  +  -450'  -  -000  555 

J273 

Cpd0  =  J(5290'  +  -000  295S70'2)  ; 


273 

and  therefore 

L  =  J  (94  -  -0790'  -  -000  85143  &-\ 
and 

'  M-WW--000  85148^ 


273  +  & 

which 

we  find 

6'     , 

0 

50 

100 

150 

c'     \ 

+  •1057 

+  •1222 

+  •1309 

+  -1344 

110.  The  fact  that  the  specific  heat  of  saturated 
steam  is  negative  was  discovered  by  Clausius  and  Ran- 
kine,  independently,  early  in  1850.  It  shows  that  when 
saturated  steam  is  compressed  adiabatically  it  becomes 
superheated,  and  that  when  it  expands  it  is  partially  con- 
densed. In  steam-engines  the  condensation  is  often  pre- 
vented by  means  of  a  'steam-jacket'  surrounding  the 
cylinder. 

All  the  conclusions  we  have  arrived  at  as  to  the 
specific  heats  of  saturated  vapours  have  been  verified 
by  the  experiments  of  Him  and  Cazin.  By  employing 
a  metal  cylinder  fitted  with  glass  at  the  ends,  the  be- 
haviour of  the  vapour  was  made  visible  to  the  eye. 
Bisulphide  of  carbon  vapour  and  steam  both  formed 


APPLICATIONS  OF  CARNOl's   PRINCIPLE.  279 

a  cloud  during  expansion  but  remained  clear  during 
compression.  Ether,  on  the  contrary,  formed  a  cloud 
during  compression  and  remained  clear  during  expansion. 

In  the  case  of  chloroform  vapour,  we  have  seen  that 
the  specific  heat  changes  sign.  Cazin  calculates  it  to  be 
zero  at  123°'48C.  In  accordance  with  this  theoretical 
result,  clouds  were  formed  during  expansion  up  to  123°  C., 
but  above  145°  C.  the  vapour  remained  perfectly  clear. 
Between  123°  C.  and  145°  C.  the  conditions  depended 
on  the  degree  of  expansion.  With  a  small  degree  of 
expansion,  there  was  no  cloud ;  but  with  more  expansion, 
a  cloud  appeared  towards  the  end  of  the  experiment, 
evidently  depending  on  the  temperature  being  reduced 
by  expansion  below  123° '48  C. 

111.  Experiments  on  latent  heat  or  on  the  volume  of 
saturated  steam  are  somewhat  uncertain,  on  account  of  the 
difficulty  of  ascertaining  when  the  vapour  is  exactly  in  the 
saturated  state  and  neither  partially  condensed  nor  super- 
heated. But  assuming  Regnault's  experimental  results 
we  have  calculated  the  following  table  referring  to  satu- 
rated steam  in  which  all  the  quantities  are  expressed  in 
the  C.G.S.  system  of  absolute  units. 

The  last  three  columns  refer  to  one  gramme  of  steam. 
To  find  s,  we  simply  add  a-  to  u,  since  u  =  s  —  a. 

The  values  of  ^4-  enable  us  to  see  the  deviations  of 

u 

saturated  steam  from  the  state  of  perfect  gas  for  which  -^ir- 
is constant. 

dw 

The  values  of  -     are  obtained  from  Clausius. 


280 


ELEMENTARY  THERMODYNAMICS. 


6' 

d 

Latent  lieat 

ps 

=  t*- 
273 

t> 

6M 

in  ergs. 

u 

0 

0 

6,134-40     120,021 

25193,860000 

209,910 

4,717,000 

5 

8,713-52  •    166,696 

25048,660000 

150,270 

4,710,000 

10 

12,222-1      229,373 

24905,070000 

108,580 

4,689,000 

15 

16,934-9 

312,054 

24760,580000 

79,347 

4,667,000 

20 

23,192-0 

420,073 

24616,030000 

58,599 

4,638,000 

25 

31,405-5 

557,431 

24471,400000 

43,900 

4,627,000 

30 

42,071-3 

732,128 

24326,680000 

33,227 

4,614,000 

35 

55,779-1 

952,166 

24181,860000 

25,397 

4,600,000 

40 

73,220-8 

1,225,550 

24032,780000 

19,610 

4,588,000 

45 

95,203-3 

1,561,610 

23891,890000 

15,300 

4,581,000 

50 

122,661 

1,973,680 

23746,720000 

12,032 

4,570,000 

55 

156,661 

2,471,100 

23601,420000 

9,551-4 

4,562,000 

60 

198,416 

3,068,540 

23455,970000 

7,644-0 

4,555,000 

65 

249,294 

3,781,990 

23310,370000 

6,163-5 

4,547,000 

70 

310,830 

4,626,140 

23164,610000 

5,007-1 

4,538,000 

75 

384,734 

5,618,310 

23018,670000 

4,097-1 

4,533,000 

80 

472,904 

6,777,180 

22872,560000 

3,374-9 

4,521,000 

85 

577,437 

8,121,400 

22726,250000 

2,798-3 

4,515,000 

90 

700,645 

9,684,360 

22579,740000 

2,331-6 

4,502,000 

95 

845,070 

11,479,300 

22433,020000 

1,954-2 

4,486,000 

100 

1,013,510 

13,530,400 

22286,080000 

1,647-1 

4,478,000 

105 

1,208,760 

15,853,400 

22138,910000 

1,396-5 

4,469,000 

110  I  1,434,080 

18,499,200 

21991,510000 

1,188-8 

4,455,000 

115   1,692,840 

21,473,100 

21843,860000 

1,017-3 

4,443,000 

120 

1,988,720 

24,816,300 

21695,950000 

874-3 

4,430,000 

125 

2,325,580 

28,551,700 

21547,780000 

754-7 

4,416,000 

130 

2,707,510 

32,716,400 

21399,330000     654-1 

4,401,000 

135 

3,138,850 

37,335,900 

21250,590000 

569-1 

4,387,000 

140 

3,624,140 

42,444,700 

21101,570000     497-1 

4,372,000 

145 

4,168,130 

48,072,400 

20952,200000     435-8 

4,358,000 

150 

4,775,810 

54,249,500 

20802,590000     383'5 

4,343,000 

155 

5,452,370 

61,020,000 

20652,630000 

338-5 

4,330,000 

160 

6,203,240 

68,402,600 

20502,330000 

299-7 

4,310,000 

165   7,033,950 

76,436,000 

20351,690000 

266-3 

4,300,000 

170  !  7,950,270 

85,156,200 

20200,410000 

237-2 

4,280,000 

175   8,958,140 

94,595,200 

20049,360000 

211-9 

4,260,000 

180  10,063,600 

104,766,000 

19897,650000 

189-9 

4,250,000 

185 

11,272,900 

115,725,000 

19745,550000 

170-6 

4,230,000 

190  12,592,500 

127,496,000 

19593,070000 

153-7 

4,220,000 

195 

14,028,600 

140,094,000 

19440,190000     138-8 

4,200,000 

200   15,588,000 

153,565,000 

19286,910000 

.  125-6 

4,180,000 

APPLICATIONS   OF   CARNOT's   PRINCIPLE.  281 

112.  We  will  now  obtain  Clausius'  expression  for  the 
energy  and  entropy  of  a  substance  of  mass  m,  existing  in 
a  state  of  stable  equilibrium,  partly  as  liquid  and  partly  as 
vapour. 

Let  us  denote  by  U0  and  <f>0  the  energy  and  entropy 
which  the  substance  possesses  when  entirely  in  the  liquid 
state  at  a  given  temperature  00  and  a  pressure  equal  to 
that  of  its  saturated  vapour  at  the  same  temperature. 
Also  let  U  and  <f>  be  the  energy  and  entropy  when  the 
temperature  is  6  and  the  mass  of  vapour  x. 

We  may  bring  the  substance  from  the  first  state  to  the 
second  by  the  following  reversible  operations : — 

(1)  Let  the  temperature  be  gradually  raised  from  #0 
to  6  without  evaporating  any  of  the  liquid,  the  pressure 
being  varied  with  the  temperature  in  such  a  way  that  at 
every  instant  it  is  exactly  equal  to  that  of  the  saturated 
vapour. 

The   heat   absorbed   will   be   m  \  Cdd\    the   increase   of 

Jet 
rd  Q 

entropy,  ml    gdO;  and  the  work  done  by  the  expansion 

of  the   liquid,  ml  p-^-dO;    where    C  and    <r   have   the 
same  meanings  as  before. 


(2)     Then   let  the  mass  x  of  vapour  be   formed   at 
constant    temperature    and    pressure.     The    increase    of 

entropy  will  be  -5-  and  the  increase  of  energy 


282  ELEMENTARY   THERMODYNAMICS. 

Hence 

•       /••  c  ,„  .  *'Z 

Je( 
and   U=U0-\ 

or      J7=  £/„+ 


The  equation  dQ  =  6d(j>  then  gives 

..  (101). 


The    last    result   may  also   be   obtained   thus.     The 
volume  v  of  the  substance  at  the  temperature  6  is 
1}  =  (in  —  x)  or  +  scs, 
=  ma  +  xu. 

Hence  in  a  small  reversible  change  of  state,  the  work  done 
on  the  substance  is 

dW=-pdv 

=  —  mp  -radd  —pd  (xu). 

The  equation  dU=  dQ  +  dW  then  gives 
dQ  =  mCdd  +  d  (xL)  -  xudp 

~d6, 

V 

=  mCdd  +  6d   X]  ,  as  before. 


113.     If  we  suppose  the  substance,  when  partly  liquid 
and  partly  saturated  vapour,  to  expand  adiabatically,  we 


APPLICATIONS  OF  CARNOT'S  PRINCIPLE.  283 

can  easily  calculate  by  means  of  the  preceding  formulae 
the  change  in  the  relative  proportions  of  the  liquid  and 
gaseous  portions  of  the  substance,  the  change  in  volume, 
and  the  work  done,  in  terms  of  the  initial  and  final 
temperatures. 

Since  the  entropy  remains  constant,  we  have,  dis- 
tinguishing the  initial  and  final  states  by  the  suffixes  ^ 
and  (2), 


or,  since  C  is  practically  constant, 

6,      a-iZj         ~  ,      B2 

'  ° 


Thus  "--*0  .........  <102> 


If  v  be  the  total  volume  of  the  substance, 

v  —  (m  —  x)  a  +  a-s 
=  ma-  +  ecu 

xL 

=  ma-  +  —  j-  . 
dp 

edd 

Hence  v,  =  ma-.,  +  7—  (  \  l  —  mC  log  7  ]  ...  (103). 

dp  \  vl  t72/ 

d0, 

It  has  already  been  shown  that  the  work,  dW,  done  on 
a  substance  in  an  indefinitely  small  change  of  state  is  not 
generally  a  complete  differential,  and  that,  in  consequence, 
the  work  W,  done  in  a  finite  "change  of  state,  generally 
depends  on  the  manner  in  which  the  change  is  effected. 
In  the  present  case,  however,  we  have  the  condition  that 


284  ELEMENTARY   THERMODYNAMICS. 

the  liquid  and  its  saturated  vapour  are  always  in  stable 
equilibrium  together.  The  path  being  specified,  the 
temperature  may  be  considered  as  the  only  independent 
variable  and  the  expression  for  dW  becomes  a  complete 
differential.  Thus 

dW  =  —  pdv  =  —pd  (ma)  —  pd  (xu), 
or,  since  the  small  quantity  ma  is  sensibly  constant, 
dW  —  —pd(xu) 


Substituting  from  equation  (99),  this  becomes 


But,  since  </>  is  constant,  we  have  from  equation  (100)  or 
(101), 

mCd0+0d  (—}  =  <), 

\  "  I 
and  therefore 

~A0 

Hence  dW=-d  (xup)  +  mCdd  +  fa  (xL)  dd. 

du 

Integrating  and  remembering  that  C  is  practically  con- 
stant, we  obtain 

W  =  -  x.,  (u,p,  -  Z2)  +  x,  (ulpl  -  ZO  +  mC  (0,  -  6,} 


+  mC(0,-01)  .........  (104). 

As  an  illustration  of  equations  (102),  (103),  and  (104), 


APPLICATIONS   OF    C  A  KNOT'S   PRINCIPLE. 


285 


the  following  table  has  been  calculated  by  Clausius,  work 
being  reckoned  in  gramme-centimetres. 
It  is  supposed  that  initially  m  grammes  of  saturated 
steam  are  contained  at  150°  C.  in  a  cylinder  impervious  to 
heat,  and  that  the  piston  is  then  slowly  drawn  out  so  that 
the  temperature  falls  and  the  steam  partially  condenses. 
In  our  formulae,  we  must  put  &\  =  m. 


0a-278 

150 

125 

100 

75 

50 

25 

& 

1 

•956 

•911 

•866 

•821 

•776 

m 

•OftQ 

•1^4. 

•1  7Q 

m 

IF 

1          1-88 

3-90 

9-23 

25-7 

88-7 

(980. 868)  m 


0      1,130,000    2,320,000  !  3,590,000  \  4,930,000     6,370,000 


114.  When  the  liquid  and  gaseous  states  exist  in 
stable  equilibrium  together,  it  is  obvious  to  the  most 
casual  observer  that  there  is  a  relation  between  the 
pressure  and  the  temperature ;  but  this  is  not  so  easy  to 
see,  without  the  assistance  of  theory,  in  the  case  of  the 
solid  and  liquid  states.  In  fact,  the  question  whether  the 
melting  point  depends  on  the  pressure  does  not  seem 
to  have  been  asked  before  it  was  answered  by  Prof. 
J.  Thomson  in  1849,  by  the  following  train  of  reasoning. 

For  the  sake  of  fixing  the  ideas,  let  the  substance 
considered  be  water,  which  contracts  in  passing  from  the 
solid  to  the  liquid  state ;  and  suppose  the  melting  point 
independent  of  the  pressure.  Let  a  large  quantity  of  it, 
partly  in  the  liquid  and  partly  in  the  solid  state,  at 


286 


ELEMENTARY   THERMODYNAMICS. 


temperature  6  and  pressure  p,  be  made  to  undergo  the 
following  cycle  of  reversible  operations  during  which 
temperature  is  kept  constantly  equal  to  6. 

(1)  Compress  the  mixture  slowly  until  the  pressure 
rises  to  p  +  dp,  without  causing  all  the  water  to  be  frozen. 
If  our  hypothesis  be  correct,  this  will  be  possible ;  for  we 
have  assumed  that  at  a  given  temperature  the  liquid  and 
the  solid  states  may  exist  together  at  all  pressures. 

(2)  Then   by   slowly   abstracting   heat,   let   a   finite 
quantity  of  the  water  be  frozen  at  constant  temperature  6 
and  constant  pressure  p  +  dp. 

(3)  Slowly  reduce  the  pressure  from  p  +  dp   to  p, 
without  melting  so  much  ice  as  was  formed  in  the  second 
operation. 

(4)  Lastly,  let  heat  be  imparted  to  the  mixture  so  as 
to  melt  some  of  the  ice  at  the  constant  temperature  9  and 
constant  pressure  p,  until  the  original  state  of  the  mixture 
is  restored. 

By  drawing  an  indicator  diagram,  it  will  be  seen  that 
there  is  a  gain  of  mechanical  work  in  the  cycle,  which,  by 
the  principle  of  the  equivalence  of  heat  and  work,  must 


APPLICATIONS   OF  CARNOT'S  PRINCIPLE.  287 

have  been  transformed  out  of  the  heat  supplied  to  the 
substance.  But  this  is  contrary  to  Carnot's  axiom,  for  all 
the  heat  gained  or  lost  by  conduction  or  radiation  may 
have  been  obtained  from  or  given  to  bodies  whose  tempe- 
rature is  uniform  and  constantly  equal  to  6. 

We  conclude  therefore  that  the  assumption  that  the 
melting  point  is  independent  of  the  pressure,  must  be 
incorrect,  and  we  infer  that  there  is  a  relation  between  p 
and  6  when  the  solid  and  liquid  forms  are  in  stable 
equilibrium  together.  In  like  manner  it  may  be  shown 
that  there  is  a  relation  between  p  and  0  when  the  solid 
and  gaseous  states  are  in  stable  equilibrium  together. 

If  we  take  two  rectangular  axes  as  axes  of  pressure 
and  temperature,  the  relations  between  p  and  9  will  be 
represented  by  three  curved  lines,  of  which  that  which 
refers  to  the  solid  and  liquid  states  is  called  by  Prof.  J. 


O  (9-273 

Thomson, the  'ice  line' ;  that  which  refers  to  the  solid  and 
gaseous  states,  the  '  hoar-frost  line  ;'  and  that  which  refers 
to  the  liquid  and  gaseous  states,  the  'steam  line.'  The 
term  'steam  line'  is  also  used  in  a  different  sense,  as  we 
have  already  seen,  as  the  name  of  the  curve  on  the  indi- 
cator diagram  which  represents  the  relation  between  the 
pressure  and  the  volume  of  unit  mass  of  saturated  vapour. 


288  ELEMENTARY   THERMODYNAMICS. 

115.  The  relation  between  the  melting  point  and 
the  pressure  is  given  by  an  equation  exactly  similar 
to  (99),  viz., 


where  L  is  the  latent  heat  of  fusion  (of  one  gramme)  in 
ergs,  u  the  difference  between  the  volumes  of  one  gramme 
of  the  substance,  when  just  at  the  melting  point,  first  in 

the  liquid  and  then  in  the  solid  state,  and  -£.  refers  to 

du 

the  ice  line. 

Hence,  since  L  is  always  positive,  if  the  substance 
expand  in  melting,  like  most  of  the  bodies  which  compose 

the  crust  of  the  earth,  Jj.  will  be  positive,  or  the  melting 

point  will  be  raised  by  increasing  the  pressure.  In  popu- 
lar language,  we  may  say  that  the  greater  pressure 
renders  the  increase  of  volume  more  difficult  and  that  in 
consequence,  a  higher  temperature  is  required  for  fusion 
than  when  the  pressure  is  less. 

The  temperature  of  the  earth  increases  rapidly  as  we 
descend,  and  at  a  moderate  depth  must  be  sufficient  to 
melt  every  substance  with  which  we  are  acquainted  when 
the  pressure  to  which  they  are  subjected  is  only  the  same 
as  that  of  the  atmosphere.  But  it  does  not  follow  that 
the  interior  of  the  earth  is  in  the  liquid  state,  because  the 
enormous  pressure  which  must  exist  there  may  be  more 
than  sufficient  to  prevent  liquefaction. 

There  are  a  few  substances,  like  water  and  cast-iron; 
which  contract  in  bulk  when  fusion  takes  place,  so  that 

u  and  -^  will  be  negative.  An  increase  of  pressure  will 
then  lower  the  melting  point,  or  assist  the  fusion. 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE.  289 

In  the  case  of  water  at  0°  C.  and  at  the  pressure  of  one 

atmo,  we  can  easily  find  the  value  of  •£.     For  the  latent 

do 

heat  is  79*25  calories,  and  the  volumes  of  one  gramme  of 
water  and  of  one  gramme  of  ice  are  1*000116  and  T087 
cubic  centimetres,  respectively.  We  therefore  have 

cW_          273  x  -086884 

dp~     79-25  x  41,539,759-8 ' 
If  dp'  be  the  pressure  dp  measured  in  atmos, 
dp  =  1,013,510  dp', 

and  therefore  ~,  =  1,013,510  d,- 

dp  dp 

273  x  -086884  x  1,013,510 

79-25  x  41,539,759-8 
=  -0073025. 

This  theoretical  result  was  verified  experimentally  by 
Sir  W.  Thomson  in  1850  in  a  very  accurate  manner.  He 
placed  a  quantity  of  water  and  lumps  of  clear  ice  in  an 
Oersted  press  which  was  fitted  with  an  ordinary  air-gauge 
to  show  the  pressure.  In  order  to  be  able  to  measure 
small  differences  of  temperature  correctly,  he  constructed 
a  thermometer  filled  with  ether-sulphide  and  enclosed  it 
in  a  larger  glass  tube  hermetically  sealed  to  protect  it 
from  the  pressure.  On  screwing  down  the  press,  the 
temperature  was  at  once  seen  to  fall,  but  the  thermometer 
returned  to  its  original  reading  when  the  pressure  was 
taken  off.  The  results  obtained  are  given  below. 

Fall  of  temperature  of  melting  point. 
Pressure. 


Observed. 


Calculated. 


8-1  atmos  -059°  C.  -059°  C. 

16-8  -129°  C.  -1227°  C. 


10 


290  ELEMENTARY   THERMODYNAMICS. 

The  small  difference  between  theory  and  observation  when 
the  pressure  is  16'8  atmos  may  be  due  entirely  to  the  fact 
that  the  theoretical  equations  refer  only  to  indefinitely 
small  changes  of  pressure. 

The  preceding  conclusions  with  respect  to  ice  have 
been  applied  to  explain  two  phenomena  which  formerly 
occasioned  much  difficulty. 

(1)  When  two  pieces  of  ice  at  the  melting  point  are 
pressed  together,  there  will  be  a  fall  of  temperature  and 
some  of  the  ice  will  be  melted.     For  since  at  a  tempera- 
ture below  the  melting  point,  ice  expands  by  heat  at 
constant   pressure,   it   follows   from   equation   (70),   that 
until  the  melting  point  is  reached,  an  adiabatic  increase 
of  pressure  will  be  attended  by  a  rise  of  temperature. 
Hence,  as  may  also  be  inferred  from  equation  (102),  when 
ice  at  the  melting  point  is  slightly  compressed  without 
loss  or  gain  of  heat,  a  small  portion  of  it  will  be  converted 
into  water  and  there  will  be  a  very  slight  fall  of  tempera- 
ture  such  that   the   water  and  ice  may  coexist  at  the 
higher  pressure  and  lower  temperature.    The  greater  part 
of  the  water  thus  formed  escapes,  so  that  the  two  pieces 
of  ice  are  left  nearly  dry.     If  the  pressure  be  then  dimin- 
ished, the  temperature  will  rise  and  the  two  pieces  of  ice 
will  be  frozen  together,  there  being  enough  moisture  left 
for  this  purpose. 

This  phenomenon  is  known  as  Regelation. 

(2)  It  was  noticed  by  Forbes  in  1842  that  a  glacier 
descends  along  its  bed  with  a  motion  like  that  of  a  very 
viscous  fluid,  such  as  tar.     His  observations  were  summed 
up  in  the  words:  'A  glacier  is  an  imperfect  fluid,  or  a 
viscous  body,  which  is  urged  down  slopes  of  a  certain 
inclination  by  the  mutual  pressure  of  its  parts.' 


APPLICATIONS   OF   CARNOT  S   PRINCIPLE. 


291 


This  is  easily  explained  by  means  of  the  principles 
which  we  have  already  laid  down;  for  a  glacier  being  a 
very  porous  mass  of  ice,  the  pressure  will  vary  greatly 
from  point  to  point ;  and  the  temperature  being  always 
about  0°  C.,  the  ice  will  melt  at  the  places  where  the  stress 
is  most  severe.  The  glacier  will  consequently  be  able  to 
take  new  forms  without  exhibiting  any  visible  rupture. 

The  behaviour  of  substances  which  expand  during 
fusion,  like  wax  and  sulphur,  was  first  examined  experi- 
mentally by  Bunsen  by  means  of  a  very  simple  and 
ingenious  apparatus.  He  took  a  glass  tube,  heated  the 
ends,  drew  them  out,  and  bent  one  of  them  round,  as  in 
the  figure.  The  thick  part  B,  of  the  tube  was  filled  with 
mercury,  the  substance  to  be  experimented  upon  intro- 
duced into  the  bent  part  C,  and  then  both  ends  of  the 


apparatus  were  hermetically  sealed  and  the  tube  fastened 
to  a  board.  The  temperature  of  the  bent  part  C  could  be 
varied  at  will  by  plunging  it  into  water  of  a  known 
temperature.  On  sinking  the  apparatus  still  deeper  in 
the  water,  the  mercury  in  B  was  heated  and  expanded, 

19—2 


292  ELEMENTARY   THERMODYNAMICS. 

causing  a  great  increase  of  pressure,  the  magnitude  of 
which  depended,  of  course,  on  the  depth  to  which  the 
part  B  was  immersed  in  the  water.  It  was  capable  of 
rising  to  above  100  atmos  and  was  very  accurately  mea- 
sured by  the  volume  of  the  air  in  the  fine  tube  A. 

The  method  of  making  the  experiment  consisted  in 
melting  the  substance  in  C  and  then,  on  allowing  the 
water  to  cool,  observing  the  temperature  and  the  pressure 
at  which  solidification  took  place.  In  order  to  show 
simultaneously  the  behaviour  of  the  substance  under  a 
pressure  of  about  one  atmo,  another  apparatus  was  con- 
structed similar  to  the  first,  but  with  the  end  A  open,  and 
fastened  on  the  same  board. 

The  two  substances  examined  by  Bunsen  were  sper- 
maceti and  paraffin,  and  the  results  obtained,  which  are 
given  below,  showed  that  the  melting  point  was  raised  by 
increasing  the  pressure.  Unfortunately,  we  do  not  possess 
sufficient  data  to  make  an  accurate  theoretical  calculation 
of  the  effects  of  pressure  on  the  melting  point. 


SPERMACETI. 

Pressure. 

Melting  point. 

1  atmo 
29  atmos 
96      „ 
141      „ 
156      „ 

47-7°  C. 
48-3°  C. 
49-7°  C. 
50-5°  C. 
50-9°  C. 

PARAFFIN. 


Pressure. 


1  atmo 
85  atmos 
100     , 


Melting  point. 

46-3°  C. 
48-9°  C. 
49-9°  C. 


APPLICATIONS   OF   CARNOT's   PRINCIPLE.  293 

116.  The  latent  heat  of  fusion  depends  on  the  tempe- 
rature of  fusion  in  the  same  way  as  the  latent  heat  of 
evaporation  depends  on  the  temperature  of  evaporation  ; 
the  relation  between  the  latent  heat  of  fusion  and  the 
temperature  being  of  the  same  form  as  equation  (98),  viz., 
dL  L 

de+c~°    J> 

where  C  and  G'  are  the  specific  heats  (in   ergs)  of  the 
solid  and  liquid,  respectively,  when  the  pressure  varies 
with  the  temperature  so  as  to  keep  the  solid  on  the  point 
of  melting  and  the  liquid  on  the  point  of  solidifying. 
To  find  C  and  (7  we  have  the  equation 

dQ=cpdd-edf0dP, 

and  therefore      C  or  C'  =  Cp  -  d  ^  ^  , 

Cp  and  -^3-  referring,  in  the  first  case,  to  the  solid  state,  in 

the  other,  to  the  liquid,  and  -^  to  the  ice  line  in  both 

cases. 

If  the  substance  considered  be  water  at  0°C.  and  at  the 

pressure  of  one  atmo,  the  value  of  ,^  may  be  taken  from 

Art.  115,  thus: 

79-25  x  41,539,759-8  dpV 
•086884  ~dfi' 


CorC'=Cp+- 


Hence  if  c,  c,  cp  be  the  equivalents  of  C,  C',  Cp,  in  calories, 


Now  the  coefficient  of  cubical  dilatation  by  heat  at  0°C. 
and  at  the  pressure  of  one  atmo,  is  —  '000  061  for  water 


294  ELEMENTARY   THERMODYNAMICS. 

and  '000153  for  ice.  Hence,  since  the  volume  of  one 
gramme  of  water  in  cubic  centimetres  is  1 '0001 16  and  the 
volume  of  one  gramme  of  ice  T087,  we  obtain 

d^=  -  -000061  for  water, 
da 

and  ^=-000166  for  ice. 

d6 

Therefore  c  =  '48  +  1514  =  "6314, 

c'=l-'0556  =  -9444. 

Consequently,  if  I  be  the  latent  heat  of  fusion  of  ice 
at  0°  C.,  in  calories,  we  obtain 
dl      /79'25 


dO~  \  273  '9 
=  "290  +  -313 
=  •603. 

For  the  relation  between  the  latent  heat  and  the  pressure, 
we  have 

dLL_dLdO 
dp      dd  dp 

=  -•603  x-2993 
=  -1804779. 

If  dp'  be  the  pressure  dp  expressed  in  atmos, 
dp  =  1,013,5 10  xdp', 

and  therefore  f ,  =  f  ^ 

dp      dp  dp 

=  1,013,510^, 
dp 

dl         1,013,510    dL 


or 


dp      41,539,759-8  dp 
=  -  '0044. 


APPLICATIONS   OF    CARNOT'S    PRINCIPLE.  295 

If  dp"  be  the  pressure  dp  in  grammes  per  square  centi- 
metre, 

dp"  =  1033-279  x  dp', 

dl       dl  dp 

and  j—r,  =  -r-,  f  „ 

dp       dp  dp 

=  -  -000  004  26. 

117.  The  change  from  the  solid  to  the  gaseous  state 
resembles  the  change  from  the  solid  to  the  liquid,  or  from 
the  liquid  to  the  gaseous  state,  and  gives  rise  to  equations 
exactly  similar;  but  owing  to  the  want  of  experimental 
data,  they  do  not  possess  much  interest.  There  is,  how- 
ever, one  general  point  of  great  importance,  which  we  will 
now  explain. 

Suppose  a  quantity  of  vapour  at  a  high  temperature 
contained  in  a  vessel  of  constant  volume  and  let  the  tem- 
perature be  slowly  reduced.  After  a  time,  the  vapour 
will  begin  to  condense  into  the  liquid  state,  and  as  the 
fall  of  temperature  goes  on,  the  amount  of  vapour  will 
become  less  and  less.  At  length,  the  liquid  will  begin  to 
freeze  and  the  three  different  states  of  aggregation  will  be 
in  stable  equilibrium  with  one  another.  We  see,  then, 
that  the  three  thermal  lines,  the  steam  line,  the  ice  line, 
and  the  hoar-frost  line,  meet  in  a  point  which  is  con- 
sequently called  the  Triple  Point. 

The  three  thermal  lines  and  the  triple  point  X,  for 
water,  are  shown  in  the  accompanying  diagram  in  which 
the  abscissa  of  a  point  is  proportional  to  0  —  273,  and  the 
ordinate  to  p  ;  OP  representing  a  pressure  of  one  atmo. 

Since  the  ice  line  is  nearly  parallel  to  Op,  the  equation 

^-,=  - -0073025,  which,  by  Art.  115,  holds  for  the  ice  line 
dp 


296 


ELEMENTARY   THERMODYNAMICS. 


at  the  point  P,  shows  that  the  temperature  of  the  triple 
point  X  lies  between  0°C.  and  -01  °C. 

"We   shall  find  it  convenient  to  distinguish  the  solid 
state  by  the  suffix (1),  the  liquid  by  the  suffix (2),  and  the 


Hoar-frost  line. 

(13).  ., 


Ice  line, 
(12). 


Steam  line. 
(23). 


-273 


gaseous  by  the  suffix  (3).  The  steam  line  may  then  be 
distinguished  by  the  suffix  (2S,,  the  ice  line  by(12),  and 
the  hoar-frost  line  by(13).  With  this  notation,  we  have, 
at  the  triple  point  X,  but  nowhere  else, 

L13  =  L12  +  Z,3, 
and  uls  =  u12  +  MS,. 

But  if  00  be  the  temperature  of  the  triple  point  X,  the 
equations 

X13  =  Uia  0odP       L       lu  Q   dp_ 

give  us  at  X, 

dp       dp      1    'Lla 


Here,  since  at  the  triple  point  X,  wls  is  so  small  in 
comparison  with  wu  and  wffl  that  we  may  take  wI3  and  w23 
to  be  equal,  we  have,  at  X, 

dp        £p       Lu 


APPLICATIONS   OF  CARNOX'S   PRINCIPLE.  297 

Consequently,  since  Z12  is  very  approximately 

79-245  x  41,539,759-8, 
we  have  at  X, 

dp  _dp__  79-245  x  41,539,759-8 
dd,3  ~  ddzl  ~~         273  x  209400 

=  57-578. 
If  dp'  be  the  pressure  dp  expressed  in  atmos, 

dp  =  1,013,510  x  dp', 
and  therefore  at  X 

H-lr'0000568' 

If  dp"  be  the  pressure  dp  in  grammes  per  square  centi- 
metre, 

dp  =  980-868  x  dp", 

and  at  X,  j~    —  -fp-  =  "0587. 

Now  the  value  of  -j§-  is  '447  at  '0°C.  and  '608  at  5°C.     It 
a  e723 

may  therefore  be  taken  to  be  '448  at  the  triple  point.     At 
this  point,  then,  we  shall  have  -=?-  =  '5067. 


PART   III. 

118.  Many  solid  substances,  such  as  the  metal  plati- 
num, have  the  power  of  absorbing  gases  to  an  appreciable 
extent  within  their  pores.  The  absorptive  powers  of  char- 
coal are  so  great  that  the  gases  absorbed  must  often  be  in 


298  ELEMENTARY   THERMODYNAMICS. 

as  dense  a  condition  as  when  they  are  liquefied  by  enormous 
pressure  and  intense  cold.  Many  liquids,  too,  possess  the 
power  of  absorbing  gases ;  and,  as  in  the  case  of  solid  ab- 
sorbents, there  are  a  few  exceptional  instances  in  which 
the  absorptive  powers  are  very  great ;  the  most  notable 
example  being  the  absorption  of  ammonia  by  water.  The 
increase  of  volume  of  a  liquid  due  to  the  absorption  of 
gases  is  generally  small  in  comparison  with  the  volume  of 
the  gases  themselves :  in  the  case  of  solid  substances,  the 
increase  of  volume  is  mostly  imperceptible. 

Again,  if  a  piece  of  salt  or  a  quantity  of  sulphuric  acid 
be  thrown  into  a  liquid,  it  will  generally  be  dissolved  and 
the  liquid  will  again  become  homogeneous.  As  more  and 
more  salt  or  acid  is  added,  the  solution  will  get  stronger 
and  stronger,  until  a  certain  state  is  reached  depending  on 
the  pressure  and  temperature  and  then,  however  much 
salt  or  acid  is  thrown  in,  it  will  be  unaffected  by  the 
liquid,  which  is  then  said  to  be  'saturated.'  At  any 
given  pressure  and  temperature,  there  is  therefore  a 
maximum  limit  to  the  strength  of  the  solution.  On  the 
other  hand,  if  we  begin  with  a  quantity  of  salt  or  sulphuric 
acid,  we  may  add  as  much  water  and,  consequently,  make 
the  solution  as  weak  as  we  please.  Lastly,  the  vapour 
emitted  by  the  solution  is  found  to  be  of  the  same 
composition  as  the  vapour  of  the  liquid  used  in  forming 
the  solution,  and  its  density,  when  in  stable  equilibrium 
with  the  solution,  is  found  to  depend  on  the  temperature 
and  on  the  strength  of  the  solution. 

Suppose  now  that  a  cylinder  fitted  with  an  air-tight 
piston,  contains  a  system  of  uniform  temperature  0,  com- 
posed of  a  given  absorbing  mass  M  and  a  given  mass  m  of 
some  other  substance,  as  a  gas,  a  salt,  a  liquid  or  its  vapour, 


APPLICATIONS   OF   CARNOT\S   PRINCIPLE.  299 

and  suppose  that  a  mass  x  of  this  substance  is  absorbed  by 
M.  Also  let  the  pressure  p  be  such  that  the  contents  of 
the  cylinder  are  in  a  state  of  stable  equilibrium.  Then  if 

we  put  jf  =  ^  we  may  obviously  take  any  two  of  the  three 

quantities  0,  p,  h,  as  independent  variables  to  define  the 
state  of  the  system  within  the  cylinder.  On  drawing  out 
the  piston,  the  unabsorbed  mass  m—x  will  generally  in- 
crease. If,  at  the  same  time,  the  temperature  be  kept 
constant,  it  will  be  necessary  to  impart  a  quantity  of  heat 
to  the  system,  depending  on  (1)  the  absorbing  part  M  +  x, 
(2)  the  unabsorbed  mass  m  —  x.  To  distinguish  between 
these  two  quantities,  we  will  suppose  the  whole  mass  m  to 
be  originally  absorbed,  or  take  x  =  m. 
Let  this  system  be  made  to  undergo  the  following 
cycle  of  reversible  operations. 

(1)  Let  the  piston  be  slowly  drawn  out,  at  the  same 
time  imparting  or  abstracting  sufficient  heat  to  keep  the 
temperature  constant,  until  a  small  quantity  of  the  sub- 
stance absorbed  is  set  free ;  and  denote  the  heat  imparted 
by  dQ  and  the  consequent  increase  of  volume  by  dv. 

(2)  Let   the  system   expand  adiabatically  until  the 
temperature  falls  to  6  —  r,  where  r  is  indefinitely  small. 

(3)  Let  the  piston  be  forced  in  at  the  constant  tem- 
perature 6  —  r  by  such  a  distance  that  the  original  state 
may  be  restored  by  a  second  adiabatic  process. 

Representing  the  cycle  on  an  indicator  diagram  by  the 
small  parallelogram  A  BCD,  let  the  isothermal  CD  be  con- 
tinued to  a  point  Q  which  represents  the  state  of  the  system 
at  the  temperature  6  —  r  when  it  first  ceases  to  be  homo- 
geneous. Then,  since  we  can  always  vary  the  second 
operation  so  as  to  make  r  positive  or  negative  at  will, 


300 


ELEMENTARY   THERMODYNAMICS. 


we  may  always  suppose  Q  to  lie  to  the  left  of  the  vertical 
line  A  P.    The  continuous  curve  CDQ  will  therefore  always 


meet  AP  in  some  point  P ;  and,  since  the  projection  of  AB 
on  the  axis  of  v  represents  dv,  the  work  done  on  the  system 
in  the  cycle  will  be  proportional  to  AP  dv. 
Again,  when  the  pressure  varies  with  the  temperature 
in  such  a  way  as  just  to  prevent  the  system  from  becoming 
heterogeneous,  the  coefficient  of  cubical  dilatation  by  heat 
will  generally  be  exceedingly  small.  The  point  Q  and  P 
will  therefore  practically  coincide  and  the  ratio  of  AN  to 
AP  may  be  taken  to  be  unity,  QN  being  perpendicular  to 
AP.  But  the  ordinate  of  A  is  proportional  to  p  and  that 

of  Q  to  p  —  T  -jj  :  hence  AN,  or  AP,  will  be  proportional 

to  -  r  -jj: ,  and  the  work  done  on  the  system  during  the 
cycle  will  be 


APPLICATIONS   OF   CARNOT's   PRINCIPLE.  301 

Carnot's  principle  then  gives 

T*£* 

dd       _r 

dQ       ~0' 

or  dQ=e^dv  ..................  (105). 

do 

If  we  put  dQ  =  Ldx,  and  dv  =  udx,  where  dx  is  the  mass  of 
the  substance  set  free  in  the  first  operation,  we  get 


(106). 


When  the  substance  absorbed  is  a  gas,  the  volume  of 
the  small  quantity  dx  will  usually  be  enormous  in  com- 
parison with  the  change  of  volume  of  the  rest  of  the  system, 
and  may,  without  sensible  error,  be  supposed  equal  to  dv. 

119.  Again,  suppose  we  have  a  pipe  in  which  there  is 
a  plug  of  charcoal,  and  let  a  steady  stream  of  gas  or  liquid 
be  flowing  slowly  through  it.  Also  let  the  temperature  of 
every  part  of  the  plug  be  kept  constant;  the  end  A,  at 
which  the  stream  enters,  being  kept  at  a  uniform  constant 
temperature  6,  and  the  other  end  B  at  a  slightly  different 
uniform  constant  temperature  6  +  dO. 


p 

J-;/Si®ftS 

>S;vSiSM 

p+dp 

When  one  gramme  of  the  gas  or  liquid  passes  through 
the  plug,  let  the  heat  evolved  at  the  end  A  of  the  plug  be 
L,  and  the  heat  absorbed  at  the  end  B,  L  +  dL.  Also  let 
the  heat  absorbed  in  the  middle  part  of  the  plug  exceed 
the  quantity  that  would  have  been  absorbed  in  the  same 
time  if  the  stream  had  not  been  flowing,  by  q. 


302  ELEMENTARY   THERMODYNAMICS. 

Then   dQ,  the   total  quantity  of  heat  absorbed  by  the 
unit  mass  of  the  stream  in  passing  through  the  plug,  will  be 


and  if  p  be  the  pressure  of  the  entering  stream  and  p  +  dp 
of  the  emergent  stream,  the  increase  of  energy,  d  U,  will  be 

dU  =  dQ-d(pv). 

Also  since  the  passage  of  the  stream  through  the  plug  may 
be  considered  reversible,  the  increase  of  entropy  of  the  unit 
mass  in  passing  through  the  middle  of  the  plug  will  be 

=jL  and  d(f>,  the  total  increase  of  entropy,  will  be 
-,  ,        L     q     L  +  dL 


0  '     V  0j 
=  q±dL_L_ 

==dQ_Lde 
Therefore  dU-0dd>  =  -d  (mti  +  5 


~d(pv)  +  ~d0 (107). 

Now  if  the  substance  which  passes  through  the  plug 
be  water,  which  is  practically  incompressible,  we  shall  have 
d  (pv)  =  0, 

and  consequently         dU-  0d(j>  =  ^dd (108). 

Again,  by  Art.  112,  we  have  for  water 
dU=Cde 


dU-0d<f>  =  0  (109). 


APPLICATIONS   OF   CARNOT's   PRINCIPLE.  303 

Comparing  equations  (108)  and  (109),  we  obtain 

L=Q. 

If  the  substance  which  passes  through  the  plug  be  air, 
or  any  of  the  perfect  gases,  pv  =  R6  and  equation  (107) 
becomes 

dU-ed(j>  =  -R  +     dd  ......  (no). 


But  for  a  perfect  gas, 

dU=Cvdd 
,  cW      v 


and  therefore       dU -  6d<t>  =  -  Rd0  +  vdp (111). 

Hence,  by  equations  (110)  and  (111), 


Comparing  equation  (112)  with  equation  (106),  and 
remembering  that  when  a  gas  is  absorbed  by  charcoal, 
the  change  in  the  volume  of  the  charcoal  is  practically  zero, 

we  see  that  -K  ,  which  refers  to  the  difference  of  pressure 

on  the  two  sides  of  the  plug,  is  equal  to  -^~-  .     The  meaning 

of  this  result,  which  has  only  been  proved  for  an  infinitesi- 
mal difference  of  temperature  between  the  two  ends  of  the 
plug,  is  that  the  mass  of  gas  absorbed  in  each  gramme  of 
charcoal  is  the  same  in  the  hotter  as  in  the  colder  part  of 
the  plug. 

120.  In  1858  an  important  formula  was  obtained  by 
Kirchhoff  which  enables  us  to  calculate  the  heat  evolved 
on  diluting  a  solution  with  water.  The  experiments  of 
Thomsen  on  the  heat  evolved  in  the  dilution  of  sulphuric 


304 


ELEMENTARY   THERMODYNAMICS. 


acid  and  the  observations  of  Babo  on  the  pressure  of  the 
vapour  emitted  by  dilute  sulphuric  acid,  furnished  results 
by  means  of  which  the  formula  was  shown  to  be  true. 

Kirchhoff  s  formula  may  be  easily  deduced  from  equa- 
tion (106)  as  follows.  Let  there  be  two  cylinders  fitted 
with  air-tight  pistons  and  joined  by  a  narrow  pipe  which 


can  be  closed  by  a  stop-cock  ;  and  suppose  the  solution 
contained  in  X  and  a  small  quantity  of  water  dw  in  Y, 
both  at  the  pressure  of  one  atmo.  Then  let  the  same 
change  of  state  be  effected  in  the  contents  of  the  cylinders 
in  two  different  ways  during  which  the  temperature  of  the 
whole  system  is  kept  constantly  equal  to  6. 

(a) 

Open  the  stopcock  and  let  the  water  in  F  be  forced 
into  X  at  the  constant  pressure  of  one  atmo.  Let  the  heat 
absorbed  be  denoted  by  dQ. 


(1)  Let  the  pressures  in  X  and  F  be  reduced  to  p  and 
P,  respectively,  so  that  the  vapour  is  just  on  the  point  of 
forming  in  both  cylinders.  Let  the  increase  of  energy  be 
called  e,. 


APPLICATIONS   OF   CARXOT  S   PRINCIPLE.  305 

Since  it  is  found  by  experiment  that  the  pressure  of 
the  saturated  vapour  emitted  by  an  aqueous  solution  is 
less  than  the  pressure  of  saturated  steam  at  the  same 
temperature  (it  clearly  could  not  be  greater),  it  follows 
that  P  is  greater  than  p. 

(2)  Let  the  water  in  Y  be  evaporated  at  the  constant 

pressure  P.     The  heat  absorbed  will  be  u6  ^dw,  which 

may  be  written  dQ1}  and  the  work  done  by  the  vapour  on 
the  piston  is  Pudw.  Since  the  error  committed  by  putting 
Pu  —  R9  is  negligible  at  low  temperatures  in  comparison 

-I         7  -p 

with  dQ1}  we  obtain,  very  nearly,  dQ1  =  Rd-^-^dw,  and 

the  increase  of  energy  becomes  dQ^  —  ROdw. 

(3)  Let  the  saturated  steam  in  Y  expand  slowly  at 
constant  temperature   until   its   pressure  is  equal  to  p. 
Since,   at    low   temperatures,   steam    behaves   somewhat 
approximately  as  a  perfect   gas,  the   increase   of  energy 
in  this  operation  will  be  very  small  compared  with  dQly 
and  may  be  omitted  altogether. 

(4)  Next  open  the  stopcock  and  then  force  the  vapour 
in  Y  slowly  into  X  at  constant  temperature.     Since  the 

Z>/3 

volume  of  the  vapour  is  originally  —  dw,  the  heat  evolved 

during  the  operation  will  be  -   -  -~dw,  which  may  be 

more  briefly  written  dQ* ;  and  since  the  pressure  never 
differs  from  p  by  more  than  an  infinitesimal  quantity,  the 
work  done  on  the  system  will  be  Rddw. 

(5)  Lastly,  let  the  pressure  be  slowly  increased  to  one 
atmo,  and   denote   the  corresponding  increase  of  energy 
by  $,. 

P.  20 


306  ELEMENTARY  THERMODYNAMICS. 

The  total  increase  of  energy  in  (6)  will  be 

dQ1  —  dQ2  +  e1  +  e2, 
or,  since  ^  is  negative  and  both  el  and  e2  very  small  in 

comparison  with  dQ, 

dQ,-dQ2. 

Again,  the  increase  of  energy  in  (a)  will  practically  be  dQ, 
since  the  work  done  may  be  neglected.     Hence 

dQ=dQ,-dQ2 (113). 

Substituting  for  dQj  and  dQz  their  values,  this  becomes 


(114). 


121.  We  will  now  give  some  numerical  illustrations 
of  equation  (106),  depending  on  the  following  elementary 
proposition.  Let  there  be  two  cylinders,  as  before,  of 
which  X  contains  a  solution  indefinitely  near  the  point 
of  saturation,  and  T  a  small  quantity  dm  of  the  substance 
absorbed,  both  at  the  same  temperature  9  and  pressure 
p ;  and  let  the  following  change  of  state  be  brought  about 
in  the  two  following  different  ways  at  the  constant 
temperature  0. 

(a) 

Open  the  stopcock  and  let  the  small  mass  dm  be  forced 
at  constant  pressure  out  of  the  cylinder  F  into  the  solution 
X.  Let  the  heat  absorbed  be  denoted  by  dQ  and  the  work 
done  on  the  system  by  dW. 

(b) 

(1)  Let  the  pressures  on  the  pistons  be  slightly 
reduced  until  the  stopcock  can  be  opened  without  break- 


APPLICATIONS   OF   CARNOT's   PRINCIPLE.  307 

iug  the  equilibrium,  and  denote  the  increase  of  energy 
by  *. 

(2)  Open  the  stopcock  and  then  slowly  force  the  mass 
dm  into  the  solution,  and  let  the  heat  absorbed  be  dQ'  and 
the  work  done  on  the  system  dW. 

(3)  Lastly,  let  the  pressure  on  the  piston  of  X  be 
brought  to  its  original  value,  the  increase  of  energy  being 
denoted  by  e2. 

Hence,  by  the  principle  of  energy, 

dW  +  dQ  =  dW  +  dQ'  +  el  +  e*. 

But  dW,  dW,  e1}  and  e2,  are  small  quantities  of  the  second 
order.  Thus 

dQ=dQ'. 

From  this  we  see  that  when  a  small  quantity  of  a  sub- 
stance is  forced,  at  constant  pressure  and  temperature,  into 
an  absorbing  solution  indefinitely  near  the  point  of  satura- 
tion, the  heat  absorbed  (or  evolved)  is  the  same  as  if  the 
same  substance  had  been  forced  at  the  same  constant 
temperature  into  a  large  solution  already  saturated  at  the 
same  pressure  and  temperature. 

In  the  first  place,  consider  the  solution  of  carbonic  acid 
by  water.  From  experiment  it  appears  that  when  the 
temperature  remains  the  same,  the  amount  of  gas  dissolved 
is  very  nearly  proportional  to  the  pressure,  up  to  a  pressure 
of  4  or  5  atmos ;  also  that  if  a  saturated  solution  be  taken 
at  the  temperature  0°C.  and  at  a  pressure  of  one  atmo,  the 

quantity  of  gas  dissolved  is  diminished  in  the  ratio    .,—  ,»» 

i.  it/0/ 

by  raising  the  temperature  to  1°C.  without  altering  the 
pressure.  In  order,  therefore,  to  preserve  the  strength  of 
the  solution  constant  when  the  temperature  rises  from  0°C. 

20—2 


308  ELEMENTARY   THERMODYNAMICS. 

to  1°C.,  the  pressure  must  be  increased  in  the  ratio 
__L^  =  1-044284.  Thus  at  0°C.  and  a  pressure  of  one 
atmo,  we  have 

0^  =  273  x  -044284  x  1013510. 
da 

The  volume  of  one  gramme  of  carbonic  acid  at  0°C.  and  a 
pressure  of  one  atmo  being  ^  cubic  centimetres,  we 

see  that  when  one  gramme  of  carbonic  acid  at  0°C.  and  a 
pressure  of  one  atmo  is  absorbed  at  this  pressure  and  tem- 
perature by  a  large  and  nearly  saturated  aqueous  solution 
of  the  same  gas,  the  heat  evolved,  in  calories,  will  be 
273  x  44-284  x  1013510_          * 

1-9774  x  41539759-8    ' 

Again,  the  masses  of  equal  volumes  of  water  and  car- 
bonic acid,  at  0°C.  and  a  pressure  of  one  atmo,  are  as  1000 
to  1'9774,  and  therefore  when  one  volume  of  carbonic  acid 
is  absorbed  as  before,  the  heat  evolved  is  sufficient  to  raise 
the  temperature  of  one  volume  of  water  by 


___ 
1000 

degrees  Centigrade. 

We  now  easily  calculate  that  the  heat  evolved  on  forc- 
ing into  solution  at  0°C.  and  a  pressure  of  one  atmo, 
1-7967  volumes  of  carbonic  acid — the  quantity  required  to 
saturate  one  volume  of  water  at  0°C. — is  sufficient  to  raise 
the  temperature  of  one  volume  of  water  by  52"C. 

At  0°C.  and  a  pressure  of  one  atmo,  carbonic  acid  may 
be  considered  a  perfect  gas,  and  it  will  be  seen  that  the 
result  we  have  obtained  relating  to  the  absorption  of  one 
gramme  of  gas,  is  practically  true  at  any  smaller  pressure ; 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE.  309 

for  example,  when  a  vessel  of  water  absorbs  carbonic  acid 
from  the  air.  In  this  case,  the  term  '  pressure,'  as  we  have 
used  it,  refers  to  the  pressure  of  the  carbonic  acid  alone, 
not  to  the  sum  of  the  pressures  of  the  carbonic  acid  and 
the  air. 

Secondly,  let  us  take  the  absorption  of  ammonia  by 
water.  In  this  case,  at  a  pressure  of  one  atmo,  the 
quantity  of  gas  absorbed  by  one  gramme  of  water  is 
•875  gramme  at  0°C.  and  '833  gramme  at  2°C.,  giving 
a  difference  of  '042  gramme.  Also  at  0°G,  the  quantity 
of  ammonia  absorbed  by  one  gramme  of  water  is  '875 
gramme  at  a  pressure  of  one  atmo  and  '906  gramme 

20 

at  800  millimetres  of  mercury,  or  —  atmos.     Thus  the 

addition  of  —  atmo  to  the  pressure  increases  the  amount 
j.y 

absorbed  by  '031  gramme,  and  therefore,  if  we  suppose 
that  when  the  pressure  does  not  differ  much  from  one 
atmo,  the  quantity  of  gas  absorbed  is  proportional  to  the 

1       42 
pressure,  the  addition  of  -^  x  —  atmo  to  the  pressure  will 

increase  the  amount  by  *042  gramme.     Hence 

-  dkp  _  273      42  x  1013510 
d0  ~   2    5        19x31       * 

Now  the  volume  of  one  gramme  of  ammonia  at  0°C.  and  a 

„  .     1000      ,  .  .  „, 

pressure  of  one  atmo,  is         -  cubic  centimetres.     We  see 

*  i  D«7  I 

then  that  the  heat  evolved  by  the  absorption  of  one 
gramme  of  ammonia,  at  0°  C.  and  a  pressure  of  one  atmo, 
by  a  large  and  nearly  saturated  solution  of  the  same  gas,  is 

273     42x1013510      1000  1 


19x31          -7-697     41539759-8 


=  3°9  caloni* 


310  ELEMENTARY   THERMODYNAMICS. 

From  this  result  we  deduce  that  when  '875  gramme  of 
ammonia — the  quantity  required  to  saturate  one  gramme 
of  water  at  0°C.  and  a  pressure  of  one  atmo — is  forced,  as 
before,  into  a  large  saturated  solution  at  this  temperature 
and  pressure,  the  heat  evolved  is  270  calories. 

The  experimental  facts  just  made  use  of  may  be  found 
in  Storer's  '  Dictionary  of  Solubilities,'  or  in  Roscoe  and 
Schorlemmer's  '  Chemistry.' 

Lastly,  let  us  consider  the  solution  of  a  salt  in  water. 
In  several  cases,  this  is  necessarily  an  irreversible  process. 
Thus  if  a  vessel  contain  a  saturated  aqueous  solution  of 
chloride  of  barium  or  sulphate  of  magnesium  together 
with  an  excess  of  the  anhydrous  salt,  a  rise  of  tempera- 
ture will  cause  part  of  the  free  salt  to  be  dissolved ;  but 
the  process  cannot  be  reversed,  however  slow  the  rise  of 
temperature  may  be ;  for  on  reducing  the  temperature, 
the  solution  deposits  a  hydrate  which  falls  peaceably 
among  the  free  anhydrous  salt.  Still,  when  the  rise  (or 
fall)  of  temperature  is  slow  enough,  the  system  will  be  in 
equilibrium  during  the  process,  and  all  the  properties  of 
reversibility  will  be  applicable. 

In  the  case  of  the  solution  of  common  salt  by  water, 
at  a  pressure  of  one  atmo,  100  parts  by  weight  of  water 
dissolve  36'7  parts  of  salt  at  0°  C.  and  1'8  parts  more  at 
100°  C.,  the  increase  being  practically  uniform.  Also  a 
solution  containing  75  grammes  of  water  and  25  grammes 
of  salt  is  of  specific  gravity  1-192,  and  therefore  its 

100 
volume  is  ,         ,  or  83'89,  cubic  centimetres.     Taking  the 

specific  gravity  of  the  anhydrous  salt  as  2'15,  the  sum  of 
the  volumes  of  the  water  and  salt  separately  would  be 
25 
~7-r  =  86'62  cubic  centimetres.     The  total  diminu- 


APPLICATIONS  OF   CARNOT'S  PRINCIPLE.  311 

tion  of  volume  is  therefore  a  little  greater  than  -J  the 
volume  of  the  salt  added.  Now  it  appears  from  experi- 
ment that  when  a  small  piece  of  salt  is  thrown  into  an 
unsaturated  solution,  the  total  diminution  of  volume  is 
nearly  independent  of  the  strength  of  the  solution.  It  is 
therefore  always  equal  to  ^  the  volume  of  the  salt  added. 

Again,  when  one  gramme  of  salt  is  thrown  into  a 
large  and  nearly  saturated  solution,  the  heat  that  must  be 
imparted  to  keep  the  temperature  constant  is  known  to 
be  8  '5  calories. 

Hence,  writing  equation  (105)  in  the  more  convenient 
form 

0_dph 

jr\      a  dhp    ,  h  dd       j 

dO=6  -j£  dv  =  --  rr  p  dv, 
*        d0  p  dgh^ 


we  find 

p  deh  _  273  x  100      1-8      1013510  I 

hdp~      36-7       X  100  X  5  x  215  X  8'5  x  41539759'S 

=  -00354. 
Putting  dp  =  ap,  so  that  a  is  small,  we  get 


Hence,  if  H  be  the  amount  of  salt  dissolved  in  a  satu- 
rated solution  at  0°  C.  and  a  pressure  of  one  atmo,  the 
amount   dissolved   at   the   same   temperature   but   at    a 
pressure  of  1  4-  a  atmos,  where  a  is  small,  will  be 
H  (1  +  -00354  a). 

122.     The  remainder  of  this  chapter  will  be  devoted 
to   capillary  phenomena.     The  systems   studied   will   be 


312 


ELEMENTARY   THERMODYNAMICS. 


supposed  so  small  in  mass  that  the  mutual  gravitation 
of  their  parts  may  be  neglected,  but  so  considerable  in 
altitude  that  the  attraction  of  the  earth  can  no  longer  be 
left  out  of  account. 

If  we  take  a  glass  tube  of  ordinary  bore,  and,  after 
heating  the  middle,  draw  it  out,  we  shall  obtain  a  tube  of 
small  bore,  known  as  a  'capillary  tube.'  On  plunging 
this  into  water,  it  will  be  seen  that  the  liquid  rises  in  the 
tube  above  its  proper  level.  On  the  other  hand,  if  the 
tube  be  plunged  into  a  trough  of  mercury,  the  mercury 
will  stand  at  a  lower  level  in  the  tube  than  elsewhere. 
This  will  explain  the  reluctance  of  mercury  to  enter  a 
fine  tube,  which,  it  will  be  remembered,  formed  a  diffi- 
culty to  be  overcome  in  making  a  thermometer. 

When  a  liquid  is  elevated  or  depressed  in  a  capillary 
tube  whose  bore  is  of  any  form  and  either  uniform  or 
variable,  it  will  often  be  possible,  by  means  of  a  slight 
touch  or  shake  by  which  no  appreciable  amount  of  work 
is  done,  to  cause  the  liquid  to  alter  its  level  in  the  tube. 
Let  us  therefore  suppose  our  system,  consisting  of  liquid, 
capillary  tube,  and  a  considerable  portion  of  the  atmo- 


sphere,  is  enclosed  in  a  large  metallic  vessel  whose  thick- 
ness is  so  great  that  its  exterior  surface  always  remains 


APPLICATIONS   OF   CARNOT's   PRINCIPLE.  313 

at  the  same  uniform  temperature  6,  whatever  actions  may 
be  going  on  inside.  Then  when  the  vessel  and  its  con- 
tents are  in  a  state  of  equilibrium  at  the  uniform  tem- 
perature 6,  let  a  slight  shake,  by  which  no  work  is  done, 
cause  the  liquid  to  change  its  level  from  P  to  Q,  where 
PQ  is  either  finite  or  infinitesimal.  Afterwards  let  a 
second  shake  be  given,  and  suppose,  if  possible,  that  the 
liquid  in  the  tube  returns  to  its  original  height  P. 
Lastly,  let  the  temperature  throughout  the  vessel  be 
made  the  same  as  at  first  without  any  further  change  in 
the  height  of  the  liquid  in  the  tube. 

Then  the  change  in  the  energy  of  the  vessel  and  its 
contents  is  zero,  and  therefore,  since  no  work  is  done  on 
it,  the  heat  absorbed  will  also  be  zero.  But  by  Carnot's 
principle,  since  the  cycle  is  irreversible  and  the  tempera- 
ture of  the  exterior  of  the  vessel  always  uniform  and 
equal  to  0,  if  Q  be  the  heat  absorbed  by  the  vessel  from 

without,  -Q  will  be  negative,  so  that  Q  must  be  negative. 

Hence  the  supposition  that  the  liquid  in  the  tube  can  be 
thus  made  to  return  from  Q  to  P,  is  absurd. 

From  the  preceding  argument  it  will  be  seen  that  if 
a  slight  shake  causes  the  liquid  to  rise  from  P  to  Q,  it 
will  be  impossible,  by  a  second  slight  shake,  to  cause  it  to 
descend  below  Q.  Thence  it  may  be  inferred  that  there 
is  some  point  0,  above  Q,  to  which  successive  slight 
shakes  will  cause  the  liquid  to  approach  nearer  and 
nearer.  If  the  liquid  stand  originally  above  the  point  0, 
a  slight  shake  will  evidently  cause  it  to  descend ;  and 
when  the  level  once  coincides  with  0,  no  shaking  will 
cause  any  further  change.  The  point  0  is  therefore  a 
position  of  stable  equilibrium  at  the  temperature  6. 


o 


314  ELEMENTARY   THERMODYNAMICS. 

It  might  perhaps  be  supposed  that  at  a  given  tempera- 
ture 6  and  a  given  pressure  of  the  atmosphere,  there  are 
several  positions  of  stable  equilibrium,  0,  0',  0",.,..  If 
this  were  the  case,  there  would  be  a  point  A  be- 
tween two  consecutive  positions  (0,  0')  of  stable 
equilibrium  such  that  if  the  liquid  stood  a  little 
above  A,  a  slight  shake  would  make  it  rise  still 
higher,  and  if  it  stood  a  little  lower,  a  slight  shake 
would  cause  it  to  descend.  However  it  is  easy  to  show, 
by  experiment,  that  there  is  but  one  position  of  stable 
equilibrium :  and  in  the  rest  of  our  discussions,  we  shall 
confine  ourselves  to  the  properties  of  this  position. 

123.  Let  a  tube  be  constructed  having  its  bore  in  its 
lower  part  very  fine,  but  in  the  upper  part  considerable. 
Then  if  we  plunge  this  tube  into  a  liquid  which  rises  in  it, 
like  water  in  a  fine  glass  tube,  it  will  be  impossible  for 
the  liquid  to  rise  into  the  wider  part  of  the  tube  so  long 
as  that  part  is  above  the  proper  level  of  the  liquid.  For, 
if  possible,  let  the  liquid  ascend  into  the  wide  part  of  the 
tube  when  that  part  of  the  tube  is  above  the  proper  level 


of  the  liquid,  and  let  the  bottom  of  the  wide  part  be  pro- 
vided with  a  spout,  like  that  of  a  pump,  so  that  the 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE.  315 

liquid  may  run  out,  as  in  the  figure.  Then  if  a  sufficient 
number  of  these  self-acting  pumps  were  employed,  they 
would  raise  enough  liquid  to  turn  a  small  water-wheel 
which  might  be  made  to  do  mechanical  work  for  us,  in 
the  usual  way,  by  means  of  a  pulley.  This  ideal  contri- 
vance might  be  in  accordance  with  the  principle  of  energy, 
for  there  might  be  enough  heat  absorbed  to  account  for 
the  mechanical  work  given  out  ;  but  we  can  easily  show 
that  it  cannot  satisfy  both  the  principle  of  energy  and 
Carnot's  principle.  For  let  such  an  ideal  system  be  con- 
structed and  let  the  whole  of  it,  except  the  shaft  and 
pulley  by  which  the  mechanical  work  is  carried  off,  be 
enclosed  in  a  large  thick  metallic  vessel,  the  exterior 
surface  of  which  is  always  at  the  same  uniform  tempera- 
ture. Also  let  the  capillary  tubes  be  fitted  with  stopcocks 
so  that  the  mechanism  can  be  stopped  or  set  in  motion  at 
will,  without  doing  work  or  imparting  or  abstracting  heat. 
Then  when  the  vessel  and  its  contents  are  in  equilibrium 
at  the  uniform  temperature  0,  let  the  stopcocks  be 
opened,  and,  after  allowing  the  mechanism  to  work  for  a 
time,  close  them  all  again  and  let  the  system  return  to 
exactly  the  same  state  as  before.  Then  since  the  change 
of  energy  is  zero,  if  W  be  the  work  done  on  the  vessel 
and  its  contents  and  Q  the  heat  absorbed  during  the 
process,  the  principle  of  energy  gives 


Also  since  the  conception  of  entropy  is  applicable  to  the 
initial  and  final  states,  we  have,  by  Carnot's  principle, 


so  that  Q  is  negative. 


316 


ELEMENTARY   THERMODYNAMICS. 


We  therefore  have  W  positive.  But,  by  hypothesis,  there 
is  a  gain  of  mechanical  work  during  the  process,  or  W  is 
negative.  We  therefore  conclude  that  the  original  sup- 
position in  accordance  with  which  the  machine  was  to  be 
constructed,  is  absurd. 

We  may  vary  the  preceding  proof  by  dispensing  with 
the  water-wheel  and  allowing  the  liquid  raised  by  the 
self-acting  pumps  to  fall  back  into  the  pool  from  which 
it  was  raised  without  doing  mechanical  work.  We  should 

then  have  the  irreconcilable  equations  Q  =  0  and  -T  <  0, 

from  which  we  may  draw  the  same  conclusion  as  before. 

Any  ideal  process,  such  as  we  have  described,  which  is 
in  contradiction  to  the  principle  of  energy,  or  to  Carnot's 
principle,  or  to  both  principles  at  once,  is  known  as  a 
'  perpetual  motion.' 

If  we  take  a  second  tube  the  bore  of  which  is  every- 


where fine  and  plunge  it  in  a  vessel  of  water  or  any  other 
liquid  which  rises  in  the  tube,  then  the  height  of  the 


APPLICATIONS  OF  CARNOT'S  PRINCIPLE.  317 

point  P  to  which  the  liquid  rises  in  the  tube,  will  be 
independent  of  the  form,  and  bore  of  the  lower  part  of  the 
tube.  For  let  AB  be  the  upper  part  of  the  tube  and  let 
us  take  another  capillary  tube  the  upper  part  A'B'  of 
which  is  exactly  similar  and  equal  to  AB ;  and  suppose,  if 
possible,  that  when  this  tube  is  plunged  into  the  same 
vessel  of  liquid  as  the  other  with  the  part  A'B'  placed 
parallel  to  and  on  the  same  level  as  AB,  the  point  P',  to 
which  the  liquid  rises  in  A'B',  is  not  in  the  same  hori- 
zontal plane  as  P.  Let  there  be  a  stopcock  at  A  and 
another  at  A,  which  can  be  turned  without  doing  work 
or  imparting  or  abstracting  heat,  and  let  these  stopcocks 
be  so  constructed  that  they  afford  the  means  of  closing 
the  downward  communication  of  the  capillary  tubes  AB, 
A'B',  and  of  causing  these  tubes  to  connect  with  one 
another  by  means  of  a  tube  A  A'  of  any  kind.  Lastly,  let 
the  whole  system  be  enclosed  in  a  metallic  vessel,  as 
before,  so  that  heat  can  only  be  absorbed  or  evolved  at 
the  uniform  constant  temperature  6.  Then  when  the 
system  is  in  equilibrium  at  the  uniform  temperature  0 
with  the  tubes  AB,  A'B'  communicating  downwards,  let 
the  stopcocks  be  turned  until  the  tubes  communicate 
only  with  one  another,  through  the  pipe  A  A.  This 
will  evidently  cause  the  liquid  to  assume  the  same  level 
in  AB  as  in  A'B',  and  there  will  therefore  be  a  flow 
through  the  pipe  AA,  which  may  either  be  used  as  a 
source  of  mechanical  work,  or  be  allowed  to  expend  itself 
wholly  in  friction.  After  a  sufficient  time,  let  the  cocks 
be  turned  back  so  that  the  system  returns  exactly  to  its 
initial  state.  Then  we  have  the  contradictory  equations 

W+Q  =  0,   W  <  0,      <  0,  or  else  Q  =  0  and     <  0.     From 


318  ELEMENTARY  THERMODYNAMICS. 

either  set   of  equations  we   conclude  that  the  points  P 
and  P'  must  be  on  the  same  level. 

Again,  if  we  take  a  tube  the  bore  of  which  is  every- 
where fine,  except  in  the  middle  (as  in  the  figure),  and 
then  plunge  it  deep  enough  in  a  liquid  which  it  depresses 


with  the  wide  part  entirely  below  the  surface,  it  is  clear 
that  the  liquid  will  fill  the  wide  part  of  the  tube  and  rise 
into  the  capillary  part  above  it.  It  may  also  be  shown 
that  the  depth  of  the  surface  of  the  liquid  column  in  the 
capillary  part  of  the  tube  is  unaffected  by  the  form  of  the 
lower  part  of  the  tube. 

It  will  now  be  seen  that  the  force  by  which  the 
liquid  is  caused  to  rise,  or  is  depressed,  in  the  capillary 
tube,  must  be  sought  for  at  the  surface  of  the  liquid 
column  in  the  tube.  Also  since  it  appears  from  experi- 
ment that  the  action  of  a  solid  on  a  liquid  is  insensible  at 
measurable  distances,  we  are  forced  to  conclude  that  the 
surface  of  the  liquid  column  in  the  tube  must  be  in  a 
state  of  tension  and  merely  held  to  the  tube  at  its  circum- 
ference. From  this  inference  we  may  anticipate  the  form 


APPLICATIONS   OF   CARNOT'S  PRINCIPLE.  319 

of  the  surface,  which,  in  the  case  of  water  in  a  glass  tube, 


is  concave  upwards,  but  in  the  case  of  mercury  in  a  glass 
tube,  convex  upwards. 

124.  Let  us  take  a  capillary  tube  whose  bore  is  uni- 
form and  in  the  form  of  a  small  circular  cylinder  of  diameter 
d,  and  let  this  tube  be  immersed  vertically  in  a  liquid 
which  rises  so  high  or  is  depressed  so  low,  that  the 
curvature  of  the  top  of  the  liquid  column  in  the  tube 
may  be  neglected  in  comparison.  Also  let  the  system 
be  enclosed  in  a  large  metallic  vessel  so  that  heat  can 
only  be  absorbed  or  evolved  at  the  constant  uniform 
temperature  0.  Then  if  we  assume  that  the  action  of 
the  tube  on  the  liquid  extends  only  to  an  exceeding 
small  distance  from  their  common  surface,  it  is  obvious 
that  when  the  system  is  in  a  state  of  equilibrium,  stable 
or  unstable,  with  the  liquid  in  the  tube  at  a  height  z 
above  its  proper  level,  with  the  temperature  uniform 
throughout  the  vessel  and  equal  to  6,  and  with  the 
practically  uniform  pressure  of  the  air  contained  within 
the  vessel  equal  to  p,  we  shall  have 

U=  U0  +  7rdzFl  (0,  p)  +  TrdFz  (0,p)} 
</>  =  0,  +  irdzf,  (0,p)  +  Trdf,  (0,p)!'" 
where    U0  and  <£0  are   independent   of  z,  and   the    four 
quantities  (F1}  -?T2,/1,/2)  depend  only  on  the  natures  and 


320  ELEMENTARY   THERMODYNAMICS. 

physical  states  of  the  tube  and  liquid  so  that  they  are 
independent  of  both  z  and  d. 

Now  if  a  slight  shake  be  given  to  the  vessel,  by  which 
the  liquid  is  caused  to  rise  in  the  tube  to  the  height 
z  +  dz,  we  shall  have,  if  p  be  the  density  of  the  liquid  and 
g  the  attraction  of  the  earth  in  dynes  on  a  mass  of  one 
gramme, 


and  dQ-ed<j><0. 

Therefore  dU—  0d$  +  :  d-gpzdz  <  0, 

so  that,  if  the  temperature  be  allowed  to  regain  its  former 
uniform  value,  the  expression 

*  *      '     (116) 


will  have  decreased  during  the  operation. 

Hence,  since  it  is  obvious  that  the  expression  must 
have  a  minimum  value,  when  that  value  is  reached  the 
system  will  be  in  a  state  of  stable  equilibrium.  To  find 
the  position  of  stable  equilibrium  at  temperature  6  we 
have  therefore  only  to  take  the  variation  of  the  expression 
(116)  on  the  supposition  that  0  is  constant,  and  then 
equate  the  variation  to  zero. 

This  result  may  also  be  obtained  thus.  When  the 
system  is  in  a  state  of  stable  equilibrium  at  the  uniform 
temperature  6,  let  z  +  dz  be  the  height  of  the  liquid  in 
the  tube  above  its  proper  level ;  and  suppose  that  when 
the  liquid  is  in  a  state  of  equilibrium  at  the  uniform 
temperature  6  with  the  liquid  only  at  the  height  z  above 
its  proper  level,  the  vessel  is  carefully  lifted  up  from  the 


APPLICATIONS   OF   CARNOT'S   PRINCIPLE.  321 

ground,  or  let  down  a  pit,  so  that  g  slightly  changes  until 
the  equilibrium  becomes  stable.  Then  let  the  vessel  be 
brought  to  its  original  position  in  such  a  way  that  the 
liquid  is  caused  to  rise  in  the  tube  in  a  reversible  manner 
to  the  height  z  +  dz.  Then  since  g  only  changes  by  an 
infinitesimal  amount  during  the  operation,  the  work  done 

on  the  vessel  and  its  contents  will  be  —  7  d?gpzdz,  and 
therefore 


Also,  since  the  operation  is  reversible, 

dQ'  -  6d<f)  =  0. 

Hence,  as  before,  if  6  remain  constant  and  the  equilibrium 
be  stable, 


Thus  if  H  be  the  value  of  z  in  the  state  of  stable  equi- 
librium at  the  temperature  0, 


0  .........  (117). 

Consequently,  since  F1  and/!  depend  only  on  the  natures 
and  physical  states  of  the  tube  and  liquid,  if  we  have 
tubes  of  the  same  substance  in  the  same  physical  state 
but  of  different  uniform  bores,  immersed  in  the  same  kind 
of  liquid,  dH  will  have  the  same  value  for  all. 

The  force  by  which  the  liquid  is  raised  or  depressed  in 
the  capillary  tube,  reckoned  positive  when  it  acts  upwards, 


21 


322  ELEMENTARY   THERMODYNAMICS. 

or,  by  equation  (117), 


which  is  equal  to  a  force  of  —  (Fl  —  df^  per  centimetre  of 
the  circumference  of  the  bore  of  the  tube. 

If  two  plates  of  the  same  nature  as  the  tube  just 
described,  be  placed  parallel  to  one  another  at  a  distance 
d,  and  immersed  vertically  in  the  same  kind  of  fluid  as 
the  tube  ;  then  if  we  consider  only  so  much  of  the  system 
as  lies  between  two  ideal  vertical  planes  at  right  angles  to 
the  plates  and  at  a  distance  of  one  centimetre  apart,  we 
shall  have,  when  the  system  is  in  a  state  of  equilibrium 
at  the  uniform  temperature  6  with  the  liquid  standing 
between  the  plates  at  a  height  z  above  its  proper  level, 

U=  Z70'  +  2zF,  (6,  p)  +  2F,  (6,  p)) 

4>  =  ^'  +  ^fl(e,p)  +  ^f,(0,p)l" 

where  U0'  and  </>„'  are  independent  of  z  and  d,  and  the 
functions  (Flt  F,,  flt  /2)  are  the  same  as  before.  Also 
when  the  equilibrium  is  stable, 

dU  -  0d<f>  +  dgpzdz  =  0; 
so  that  if  h  be  the  value  of  z  in  that  case, 

2(Fi-Ofi)  +  dgph  =  Q  ............  (119). 

Comparing  equations  (117)  and  (119),  we  see  that 

H=2h, 

or  that  the  liquid  is  raised  or  depressed  between  the  two 
plates  exactly  half  as  much  as  in  the  tube. 

125.  The  earliest  application  of  thermodynamics  to 
capillary  phenomena  was  made  by  Sir  W.  Thomson  in 
1870. 


APPLICATIONS   OF   CARNOT S   PRINCIPLE. 


323 


If  a  fine  glass  tube  A,  open  at  both  ends,  be  plunged 
with  its  lower  extremity  in  a  sheet  of  water,  the  water 
will  be  seen  at  once  to  ascend  the  tube,  and  it  is  obvious 
that  in  the  state  of  stable  equilibrium,  the  equilibrium 
will  not  be  disturbed  by  closing  the  lower  end  of  the  tube. 
Hence  we  infer  that  if  a  second  tube  B,  exactly  similar 


and  equal  to  A,  but  with  its  lower  end  closed,  be  held 
parallel  to  and  on  the  same  level,  as  A,  equilibrium  will 
not  be  completely  established,  if  there  is  any  vapour  in 
the  air,  until  the  water  is  at  the  same  level  in  B  as  in  A. 
The  tube  B  must  therefore  have  the  power  of  condensing 
the  aqueous  vapour  of  the  atmosphere.  As  this  process 
goes  on,  the  quantity  of  water  inside  the  tube  B  will 
increase  and  the  density  of  the  vapour  on  its  top  become 
less  and  less,  until  at  last  the  state  of  equilibrium  is 
attained. 

If  p  be  the  pressure  and  p  the  density  of  the  vapour 
in  the  air  just  over  the  level  sheet  of  water,  and  p  the 
pressure  of  the  vapour  at  the  height  h  of  the  water 
column,  we  have 


since  the  variations  of  the  density  in  the  height  h 
small  in  comparison  with  p. 

21—2 


are 


324  ELEMENTARY   THERMODYNAMICS. 


Hence  p  =p    l  -  . 

Now  if  the  temperature  be  10°  C.,  p  =  12,222  and 

_JL 

p  ~  108,581  ' 
Thus,  since  we  may  take  #=981,  we  obtain,  at  10°  C., 

P  -^  I1      1,352,800,1  ' 

Also  in  a  tube  whose  radius  is  the  thousandth  of  a 
millimetre,  it  is  calculated  that  water  would  rise  to  a 
height  of  about  13  metres  above  the  plane  level.  Con- 
sequently the  equilibrium  pressure  of  the  aqueous  vapour 
in  the  tube  will  be  less  than  just  over  the  plane  surface  of 
the  water  by  about  the  thousandth  part. 

Sir  W.  Thomson  thinks  it  probable  that  the  moisture 
which  vegetable  substances,  such  as  flannel,  cotton,  etc., 
acquire  from  the  air  at  temperatures  far  above  the  '  dew- 
point  '  —  that  is,  far  above  the  temperature  at  which,  in  a 
given  state  of  the  air,  vapour  begins  to  be  deposited  on 
plane  surfaces  —  may  be  accounted  for  by  the  condensation 
of  vapour  in  the  minute  cells  of  the  substance. 

This  article  will  also  help  us  to  understand  the  fact, 
which  we  have  already  noticed,  that  when  the  bulb  of  a 
thermometer  is  wrapped  with  flannel  and  then  breathed 
upon,  the  temperature  of  the  bulb  rises  so  much  above 
that  of  the  breath. 


CHAPTER  V. 

THE   THERMODYNAMIC   POTENTIAL. 

126.  THE  object  of  the  present  chapter  is  to  find 
a  simple  method,  corresponding  to  the  method  of  the 
Potential  in  Statics,  which  shall  determine  whether  the 
equilibrium  of  a  system  is  stable  or  unstable,  and  give 
the  properties  of  stable  equilibrium. 

The  first  attempt  of  this  kind  was  made  by  Berthelot. 
The  rule  which  he  proposed  under  the  name  of  the 
'  principle  of  maximum  work '  was  very  simple. 
Thus  if  a  system  be  in  a  state  of  equilibrium,  the  tempera- 
ture will  be  uniform  throughout.  If  now,  without  any 
interference  from  other  systems,  any  disturbance  or  chemi- 
cal reaction  takes  place,  the  temperature  will  be  rendered 
variable  but  ultimately  it  will  again  become  uniform. 
Berthelot  then  supposes  that  the  final  temperature  is 
higher  than  the  first,  and  that,  consequently,  if  the 
temperature  fall  to  its  original  value  by  conduction  or 
radiation,  the  system  must  give  out  a  positive  quantity  of 
heat.  This  assumption  may  be  more  simply  expressed  by 
saying  that  every  chemical  reaction  tends  to  make  the 


326  ELEMENTARY   THERMODYNAMICS. 

system  assume  that  state  in  the  production  of  which  it 
liberates  most  heat.  Or  again,  since  the  system  is  not 
at  liberty  to  receive  or  lose  energy  except  in  the  form  of 
heat,  we  may  say  that  the  energy  of  the  system  tends 
to  a  minimum  value. 

Berthelot's  rule  can  be  easily  applied  and  its  results 
are  often  in  satisfactory  agreement  with  experiment. 
There  are,  however,  a  few  common  cases  in  which  it 
altogether  fails.  For  example,  if  salt  and  snow  be  mixed 
together,  there  is  a  considerable  fall  of  temperature,  and 
if  the  original  temperature  be  restored,  there  will  be  an 
absorption  of  heat  and  an  increase  of  energy. 

A  correct  test  of  stability  is  afforded  by  the  theory  of 
entropy.  We  have  already  seen  that  when  any  material 
system  is  prevented  from  receiving  or  losing  heat,  its 
entropy  is  constantly  increasing  except  when  the  system 
is  in  a  state  of  stable  or  unstable  equilibrium.  The 
equilibrium  will  therefore  be  stable  when  the  entropy 
of  the  system  is  a  maximum.  For  instance,  if  the  system 
be  in  equilibrium  in  a  state  P  for  which  the  entropy  is  a 
maximum,  it  will  be  impossible  for  the  system  to  pass 
from  the  state  P  to  any  neighbouring  state  Q,  because 
this  would  require  the  entropy  to  be  greater  in  the  state 
Q  than  in  the  state  P,  since  the  path  from  P  to  Q  can 
never  be  strictly  reversible  or  non-frictional. 

The  method  of  entropy  is  not  of  much  practical  value, 
on  account  of  the  difficulty  of  making  the  calculations. 
A  different  test  of  stability  is  therefore  used  which  is 
much  simpler  in  application,  because  it  supposes  the 
temperature  constant.  The  properties  of  the  functions 
which  are  employed  for  this  purpose  appear  to  have  been 
first  investigated  by  Massieu  in  1869,  but  he  did  not 


THE   THERMODYNAMIC    POTENTIAL.  327 

examine  whether  they  furnished  a  test  of  stability.  This 
was  done  independently  by  Prof.  W.  Gibbs  in  1875  and 
by  Dr  Helmholtz  in  1882.  It  is  worthy  of  notice  that 
the  earlier  Thermodynamicists  have  had  no  part  in  dis- 
covering the  method. 

For  the  sake  of  simplicity,  we  shall  suppose  every  part 
of  the  bodies  or  systems  which  we  consider  to  be  readily 
permeable  to  heat,  so  that,  in  a  state  of  equilibrium,  the 
temperature  will  be  uniform  throughout.  When  such  a 
body  or  system  undergoes  a  small  reversible  modification, 
we  have 


dQ  =  0d<}>          ' 

and  therefore  d  W  =  d  U  -  Sd$. 

Hence,  if  the  temperature  remain  constant  during  the 
process, 

dw=d(U-e<i>). 

If  we  put  f  for  U—0<J),  the  preceding  equation  becomes 

dW=df. 

We  see  then  that  if  a  body  undergoes  a  reversible  opera- 
tion during  which  the  temperature  is  kept  constant,  the 
function  f  ",  which  depends  only  on  the  state  of  the  body, 
will  be  increased  by  the  amount  of  work  done  on  the 
body,  or  decreased  by  the  amount  of  work  done  by  the 
body.  For  this  reason,  the  function  ZF  is  called  by  Helm- 
holtz the  'free  energy/  being  that  part  of  the  energy 
which  is  convertible  into  mechanical  work  at  the  constant 
temperature  0.  In  like  manner,  0<f>  is  called  the  '  bound 
energy,'  because  it  is  not  convertible  into  work.  No 
further  reference  will  be  made  to  this  nomenclature,  and 
it  should  be  noticed  that  it  has  no  connection  whatever, 


328  ELEMENTARY   THERMODYNAMICS. 

beyond  a  similarity  of  sound,  with  our  conceptions  of  free 
and  bound  etherial  energy;  the  names  of  which  have 
been  suggested  to  us  by  reading  Lodge's  '  Modern  views 
of  Electricity.' 

When  the  only  external  forces  to  which  the  body  is 
subjected  consist  of  a  uniform  and  constant  normal 
pressure,  p,  on  the  surface,  dW  =  —  pdv,  so  that  if  we 
write  <£  for  f  +pv,  or  U—dfy+pv,  the  preceding  result 
becomes 

d®  =  0. 

If,  under  these  conditions,  v  be  also  constant,  then 
df  =  Q. 

Let  us  now  suppose  that  the  system  is  in  equilibrium 
in  any  state  A  at  any  uniform  temperature  6,  Then  the 
equilibrium  in  the  state  A  will  be  unstable  if  a  slight 
shake  or  touch,  by  which  no  perceptible  change  is  made 
in  the  system,  causes  the  equilibrium  to  be  broken  in 
consequence  of  which  the  system  rushes  into  some  other 
state  P.  Hence  clearly  the  equilibrium  in  the  state  A 
will  be  stable  if  every  spontaneous  change  of  state,  like 
AP,  is  impossible. 

In  order  to  express  this  condition  mathematically,  let 
us  suppose  the  system  enclosed  in  a  vessel  which  is  so 
rigid  that  no  mechanical  work  can  be  obtained  from  it, 
and  so  slow  a  conductor  of  heat,  especially  at  its  inner 
surface,  that  the  temperature  of  its  exterior  surface  is 
always  uniform  and  equal  to  6,  whatever  processes  may 
be  going  on  inside.  Let  us  further  suppose  the  vessel  to 
be  such  that  the  same  external  forces  act  on  the  system 
when  it  is  inside  the  vessel  as  when  the  vessel  is  not 
employed.  Also  let  the  whole  of  the  vessel  as  well  as  its 


THE   THERMODYNAMIC   POTENTIAL.  329 

contents  be  originally  at  the  uniform  temperature  0,  and 
after  the  spontaneous  change  of  state  from  A  to  P,  let  us 
wait  till  the  vessel  and  its  contents  are  of  the  same  uni- 
form temperature  as  before,  and  call  the  new  state  of  the 
system  within  the  vessel  B. 

Then  since  there  is  no  change  in  the  state  of  the  contain- 
ing vessel,  and  since  the  total  quantity  of  heat  absorbed 
by  it  during  the  interval  is  zero,  the  heat  absorbed  by 
the  system  within  the  vessel  will  be  equal  to  the  heat 
absorbed  by  the  vessel  at  its  exterior  surface.  Hence  if 
AQ  be  the  heat  absorbed  by  the  system  during  the  change 
of  state  AB,  the  magnitude  of  which  may  be  finite,  and 
A<£  the  increase  of  entropy,  we  shall  have 


or  AQ  <  0A</>. 

But  if  At/"  be  the  increase  of  energy  of  the  system  and 

AW  the  work  done  on  it,  the  principle  of  energy  gives 


Hence  AJ7-  AW<  0A<£, 

or  AZ7-0A0<ATP. 

Thus  since  6  has  the  same  values  in  both'  of  the  states 

A,B, 


that  is,  AF-ATT<0. 

Now  if  the  external  conditions  be  such  that  AW  is  a 
complete  differential,  the  value  of  AJF  —  AW  will  depend 
only  on  the  states  A  and  B,  and  we  shall  be  able  to  draw 
some  remarkable  conclusions.  This  will  happen  in  two 
very  simple  and  important  cases  to  which  we  shall 


330  ELEMENTARY   THERMODYNAMICS. 

restrict  ourselves  for  the  future  in  discussing  questions 
of  stability.  In  both  of  these  cases,  the  external  force 
is  a  uniform  normal  pressure  on  the  surface  :  in  the 
first  case,  the  volume  is  also  constant,  and  in  the  second, 
the  surface  pressure  is  constant  as  well  as  uniform. 
When  the  volume  is  constant,  A  W  =  0,  and  therefore 
AiF  must  always  be  negative.  When  the  pressure  is  con- 
stant, A  W  =  —  p  AV  =  —  A  (  pv),  and  therefore  A  (JF  +  pv), 
or  A<£>,  must  always  be  negative.  Hence  when  the 
function  f  ',  or  the  function  <&,  attains  a  minimum  value 
corresponding  to  any  given  temperature,  the  change  of 
state  from  A  to  B  will  cease  to  be  possible,  and  conse- 
quently also,  the  spontaneous  change  from  A  to  P.  In 
other  words,  A  will  be  a  state  of  stable  equilibrium. 

This  may  also  be  shown  in  a  different  way.  For  it 
will  be  possible,  by  violating  the  conditions  of  volume  or 
pressure,  to  bring  the  system  from  A  to  B  by  a  reversible 
operation  during  which  the  temperature  is  constantly 
uniform  and  equal  to  6.  If  in  this  case,  AQ'  be  the 
amount  of  heat  absorbed,  we  shall  have 


Hence,  when  the  volume  is  kept  constant, 


=  AQ  -  <9A</> 

=  AQ  -  AQ', 
and  when  the  pressure  is  kept  constant, 

A<£  =  AQ  -  AQ'. 

Now  when  A,  B  are  states  of  unstable  equilibrium  near 
together,  the  amount  of  work  done  on  the  system  during 
the  reversible  path  will  clearly  differ  by  a  finite  quantity 


THE  THERMODYNAMIC    POTENTIAL.  331 

from  the  work  actually  done  when  the  system  passes  of 
itself  from  A  to  B  under  the  given  conditions  of  volume 
or  pressure.  The  heat  absorbed  will  therefore  differ  by  a 
finite  quantity  in  the  two  cases  and  the  variations  of  f 
and  <I>  will  consequently  be  finite.  But  when  A,  B  are 
states  of  stable  equilibrium,  the  difference  in  the  work 
done  or  in  the  heat  absorbed,  will  be  a  small  quantity  of 
the  second  order,  which  may  be  neglected.  A  state  of 
stable  equilibrium  will  therefore  make  df  =  0  when  the 
volume  is  constant,  and  d<&  =  0  when  the  pressure  is 
constant,  the  temperature  being  supposed  constant  in  effect- 
ing the  differentiation. 

Since  U  and  <£  both  contain  an  arbitrary  constant,  the 
functions  f  and  <E>  will  each  contain  an  arbitrary  term  of 
the  form  a  +  (36,  where  a  and  /3  are  constants.  But  this 
does  not  affect  the  condition  of  stability,  because  the 
temperature  remains  constant  in  taking  the  variations  of 
f  and  <&. 

On  account  of  the  important  properties  just  obtained, 
which  include  the  '  energy  test '  of  stability  in  advanced 
abstract  dynamics,  the  function  f  is  called  by  Duhem  the 
'  Thermodynamic  Potential  at  constant  volume,'  and  <E>  the 
'  Thermodynamic  Potential  at  constant  pressure.' 

127.  To  obtain  the  formulae  of  M.  Massieu,  let  us 
suppose  the  state  of  the  substance  to  be  completely 
defined  in  equilibrium  by  the  temperature  and  volume,  so 
that  0  and  v  may  be  taken  as  independent  variables. 
Then  for  a  change  from  the  state  (d,  v)  to  the  state 
(0  +  dd,  v  +  dv),  we  have 

$F=dU-ed$-$dO- 

This  result  will  not  be  true  if  0  and  v  are  not  sufficient 


332  ELEMENTARY   THERMODYNAMICS. 

to  define  completely  the  state  of  the  system ;  for  if  the 
change  of  <f>  corresponding  to  dd  and  dv  were  finite,  the 
increment  of  0$  would  not  be  0A<£  +  $dO.  If  the  pressure 
on  the  surface  during  the  change  of  state  be  uniform  and 
equal  to  p,  we  have 

dU=dQ-pdv, 

and  therefore  df=dQ-  0d<j>  -  $dd  -  pdv. 

Now  the  preceding  process  will  not  generally  be  reversible, 
but  if  the  equilibrium  be  stable,  the  same  small  change  of 
state  may  be  brought  about  at  constant  temperature  0  by 
a  reversible  process  in  which  the  work  done  on  the  system 
only  differs  from  —  pdv  by  a  small  quantity  of  the  second 
order.  We  shall  therefore  have  dQ  =  0d(f>, 

and  consequently,     df  =  —  faW  —  pdv, 


d20). 


By  means  of  these  results  we  can  easily  express  every 
quantity  referring  to  the  substance  in  terms  of  (0,  v,  F) 
and  the  differential  coefficients  of  f  with  respect  to  6  and 
v.  Thus  we  have 


,...(121). 


7     JJ  fj2f 

Also  Cv=  -^-  =  _0_- (122), 

7 

and  since  dQ 


THE   THERMODYNAM1C   POTENTIAL.  333 


we  obtain        Cp  =  Cv  +  6       - 


—  a    j 
dep 

dv 


Again,  if  .Kg  be  the  isothermal  compressibility  and  e 
the  coefficient  of  cubical  dilatation  by  heat  at  constant 
pressure, 


dep 
dv 


When  6  and  p  can  be  taken  as  independent  variables, 
we  have 

d®  =  dU-6d<f>-  <f>d0  +pdv  +  vdp, 


334  ELEMENTARY   THERMODYNAMICS. 

or,  since  dU=dQ—pdv  and  in  a  state  of  stable  equili- 
brium dQ  =  6d(f>, 

d<&  =  —  <f>d6  +  vdp, 

and  therefore  -rz  =  — 


and 


Hence  also  U=^-6^-p   ^      (127), 

d&      r  dp 

„      dJJ        d»v 


dp2 
Again  K6  =  -l-df- 


(130), 


THE  THERMODYNAMIC   POTENTIAL.  335 

1  dvv 


128.     As  an  illustration,  we  will  find  fF  and  <f>  for  unit 
mass  of  ideal  perfect  gas  satisfying  the  relation  pv  =  R6. 

Since  we  have       dU=  Cvd0 


j,      n  dd     pdv 
we  find  «9  =  0^  -^  +  ~5~  > 

-,,      n  dd      p  dv 

or  d<j>  =  Cv-+R- 


Hence  CT-  U,=  CV(0-60\ 

and  0  -  <£„  =  ^o  log  £  +  R  log  - 

"o  "0 


V  »0 

since  -  =  -/r      • 

V0       ^o^ 

Thus 


132). 


336  ELEMENTARY   THERMODYNAMICS. 

From  these   equations,  taking  6  and  v  as  independent 
variables,  we  get 


dw  _   ne  _ 

~fo~~^~  =  ~p- 

Taking  6  and  p  as  independent  variables, 


d®  _R0_ 

dp  ~  p  ~ 

Now  the  equation  pv  =  R0  shows  that  at  a  given 
temperature  and  pressure,  the  value  of  R  for  different 
perfect  gases  is  proportional  to  v.  Hence  if  D  be  the 
density  of  the  gas,  that  is,  the  mass  contained  in  a  unit  of 
volume,  at  a  given  temperature  and  pressure,  say  at  0°  C. 
and  at  a  pressure  of  one  atnio,  we  have 

7?        k 

R  =  T>> 

where  &  is  a  constant  which  has  the  same  value  for  all 
perfect  gases.     Equations  (132)  therefore  become 


(133). 


129.  The  methods  of  entropy  and  of  the  thermo- 
dynamic  potential  are  combined  in  a  beautiful  geometrical 
construction  due  to  Prof.  W.  Gibbs,  which  does  not  seem 


THE   THERMODYXAMIC   POTENTIAL. 


337 


to  have  obtained  the  attention  it  appears  to  deserve. 
This,  no  doubt,  is  owing  to  the  fact  that  W.  Gibbs'  works 
are  practically  inaccessible  to  European  readers,  the  only 
abstract  with  which  we  are  acquainted — in  addition,  of 
course,  to  the  very  brief  notice  in  Maxwell's  '  Theory  of 
Heat ' — being  given  in  Duhem's  treatise  on  the  Thermo- 
dynamic  Potential. 

In  Gibbs'  geometrical  method,  \vhich  supposes  the 
substance  to  be  acted  on  by  no  external  forces  except  a 
pressure  on  the  surface  which  becomes  uniform  in  a  state 


of  equilibrium,  we  take  three  rectangular  axes :  the  axis  of 
x  representing  the  entropy,  the  axis  of  y  the  volume  and 
the  axis  of  z  the  energy.  The  different  states  of  equili- 
brium of  the  substance  will  then  be  represented  by  a  surface. 
Now  at  any  point  (x,  y,  z)  of  the  surface,  the  direction- 
cosines  of  the  normal  are  known  to  be  proportional  to 

fdyZ      dxz 

(duo'    ~dy' 

fd   TT       /I    J 

that  is,  to  I  —. 


dv 


-  1 


22 


338 


ELEMENTARY   THERMODYNAMICS. 


But  since  any  line  drawn  on  the  surface   represents   a 
reversible,  or,  at  least,  an  equilibrium  path,  we  have 


Thus 


dvU     a      .d^U 
-     =  ®  and      ~  =  ~ 


The  direction-cosines  of  the  normal  are  therefore  propor- 
tional to  (6,  —p,  —  l).  Hence  any  two  points  of  the  surface 
at  which  the  tangent  planes  are  parallel  correspond  to  two 
states  in  which  the  substance  is  in  equilibrium  at  the 
same  temperature  and  pressure. 

We  are  now  prepared  to  examine  whether  a  state  of 
equilibrium  is  stable.  For  let  the  substance  to  be  con- 
sidered be  contained  in  a  cylinder  fitted  with  a  smooth 
air-tight  piston  and  placed  within  a  closed  vessel  of 


immense  size,  of  invariable  form  and  volume,  covered  with 
the  best  kind  of  non-conducting  material.  Then  let  the 
large  vessel  be  filled  with  some  fluid  or  gas  whose  state  of 


THE   THERMODYNAMIC   POTENTIAL.  339 

equilibrium  is  always  stable.  If  the  fluid  be  light  and 
compressible,  like  air,  we  shall  have  the  case  of  the 
thermodynamic  potential  at  constant  pressure :  if  the 
fluid  be  incompressible,  we  obtain  the  case  of  the  thermo- 
dynamic potential  at  constant  volume.  In  what  follows, 
we  suppose  the  fluid  to  be  air. 

If,  under  these  conditions,  the  substance  can  exist  in 
any  two  different  states  of  equilibrium  whatever,  it  is 
obvious  that  the  volume  and  the  energy  of  the  whole 
system  will  have  the  same  value  in  both  cases:  also,  on 
account  of  the  immense  size  of  the  larger  vessel,  that  the 
pressure  and  the  temperature  will  have  the  same  uniform 
values  in  both  states. 

The  equilibrium  in  a  state  A  will  be  stable,  under  the 
given  conditions  of  pressure,  if  the  system  is  unable,  of 
itself,  to  pass  into  any  other  state  P.  Now  the  tempe- 
rature in  the  state  P  will  not  generally  be  the  same,  at 
first,  as  in  the  state  A  ;  but  by  allowing  the  system  to 
stand  long  enough  and,  if  necessary,  removing  part  of  the 
non-conducting  covering  of  the  cylinder,  it  will  always  be 
possible,  without  violating  the  conditions  of  the  system,  to 
bring  the  system  to  a  third  state  B  in  which  the  tempe- 
rature has  its  original  uniform  value.  We  have,  therefore, 
only  to  determine  whether  the  system  is  able,  of  itself,  to 
pass  from  the  state  A  to  any  other  state  B  at  the  same 
temperature  and  pressure  as  A. 

Since  the  system  is  unable  to  receive  or  lose  heat  on 
account  of  the  non-conducting  covering,  the  substance 
will  be  unable,  of  itself,  to  pass  from  A  to  B  if  the 
entropy  of  the  whole  system  be  greater  in  the  state  A 
than  in  the  state  B.  To  exhibit  this  condition  geo- 
metrically, let  the  entropy,  volume  and  energy  of  the  fluid 

22—2 


340 


ELEMENTARY   THERMODYNAMICS. 


which  surrounds  the  cylinder  be  represented  by  three 
rectangular  axes  drawn  in  opposite  directions  to  the 
former  set  of  axes.  Then  since  the  quantity  of  fluid  is 
very  great,  all  its  states  of  equilibrium  will  be  represented 
by  a  plane  surface.  Also,  if  we  choose  the  origins  of 
coordinates  such  that  when  the  substance  within  the 
cyclinder  is  in  equilibrium  with  the  surrounding  fluid  at 
any  given  pressure  and  temperature,  the  corresponding 
states  are  represented  by  two  points  lying  on  a  line 
parallel  to  0(f>,  it  is  clear  that  any  other  state  of  equilibrium 
will  be  represented  by  two  points  also  lying  on  a  line 
parallel  to  0(f>. 

The  accompanying  figure  is  supposed  to  be  a  view  of 
the  surface  as  seen  by  an  observer  at  a  great  distance  on 


O  0 

the  axis  Ov.  The  plane  which  represents  the  states  of 
the  external  fluid  is  intersected  in  the  points  X,  Y,  by 
lines  through  A  and  B  parallel  to  0</> ;  and  A  M,  XN  are 
planes  parallel  to  the  plane  UOv. 

The  increase  of  the  entropy  of  the  substance  within 
the  cylinder  will  therefore  be  proportional  to  BM  and  the 
decrease  of  the  entropy  of  the  rest  of  the  system  to  NT. 


THE   THERMODYXAMIC   POTENTIAL. 


341 


The  total  increase  of  entropy  is  therefore  proportional  to 
MB  -  NY  =  MB  +  MY  -  NM  =BY-AX=  BL. 

If,  therefore,  the  point  L  be  to  the  right  of  the  point 
B,  the  substance  will  be  unable  to  pass  from  the  state  A 
to  the  state  B  and  the  point  A  represents  a  stable  con- 


dition of  the  substance  within  the  cylinder.  On  the  other 
hand,  if  L  be  to  the  left  of  B,  the  point  A  will  represent 
a  state  of  the  substance  which,  if  possible  for  an  instant,  is 
essentially  unstable  and  cannot  be  permanent. 

Instances  of  unstable  states  occur  '  when  a  liquid  not 
in  presence  of  its  vapour  is  heated  above  its  boiling  point, 
and  also  when  a  liquid  is  cooled  below  its  freezing  point, 
or  when  a  solution  of  a  salt  or  a  gas  becomes  super- 
saturated. 

In  the  first  of  these  cases,  the  contact  of  the  smallest 
quantity  of  vapour  will  produce  an  explosive  evaporation ; 
in  the  second,  the  contact  of  ice  will  produce  explosive 
freezing ;  in  the  third,  a  crystal  of  the  salt  will  produce 
explosive  crystallization ;  and  in  the  fourth,  a  bubble  of 
any  gas  will  produce  explosive  effervescence.' 

Again,  when  the  surface  touches  a  tangent  plane  in 


342  ELEMENTARY  THERMODYNAMICS. 

two  points  A,  B,  and  is  to  the  left  of  it  everywhere  else, 
portions  of  the  substance  can  permanently  coexist  at  the 

A 


same  temperature  and  pressure  in  the  two  states  corre- 
sponding to  the  points  A,  B,  and  it  will  be  possible  to 
pass  from  one  state  to  the  other  by  a  reversible  operation 
during  which  the  temperature  and  pressure  remain  con- 
stant. 

If  a  mass  x  in  the  state  A  be  in  equilibrium  with  a 
mass  1  —  x  in  the  state  B,  the  corresponding  point  in  the 
diagram  will  be  a  point  in  the  line  AB,  coinciding  with 
the  centre  of  gravity  of  a  mass  x  placed  at  A  and  a  mass 
1  —  x  placed  at  B.  Hence,  since  x  may  have  all  possible 
values  between  0  and  1,  every  point  in  the  limited  line 
AB  represents  a  condition  of  the  substance  when  it  is 
partly  in  the  state  A  and  partly  in  the  state  B.  If  now 
the  tangent  plane  roll  on  the  'primitive'  surface  repre- 
senting the  state  of  the  substance  when  homogeneous, 
the  locus  of  the  line  AB  will  be  another  surface,  called 
the  '  secondary '  surface,  which  represents  the  condition  of 
the  substance  when  part  is  in  one  state  and  part  in 
another. 

For  the  sake  of  fixing  the  ideas,  let  the  points  A,  B, 
belong  to  the  liquid  and  gaseous  states,  respectively. 
Then  there  will  be  two  other  rolling  planes  corresponding 
to  the  coexistence  of  the  solid  with  the  liquid,  and  of  the 


THE   THERMODYNAMIC   POTENTIAL.  343 

solid  with  the  gaseous,  states.  If  the  first  of  these  touch 
the  surface  in  the  same  point  A0  as  the  rolling  plane  AB, 
the  two  planes  will  then  coincide,  since  the  surface  is 
known  to  be  continuous  and  to  possess  only  one  tangent 
plane,  at  A0.  The  tangent  plane  which  touches  the 
surface  in  the  point  A0  will  therefore  touch  in  two  other 
points  B0,  C0.  In  this  case,  the  three  rolling  planes 
coincide :  the  physical  interpretation  being  that  the 
substance  can  then  exist  simultaneously  in  the  three 
forms  of  solid,  liquid  and  gaseous. 

The  three  developable  surfaces  obtained  by  means  of 
the  three  rolling  planes,  together  with  the  plane  triangle 
AoBoCo,  which  corresponds  to  the  'triple  point,'  constitute 
what  Prof.  Gibbs  calls  the  '  Surface  of  Dissipated  Energy.' 


CHAPTER  VI. 

APPLICATIONS    OF   THE   THERMODYNAMIC    POTENTIAL. 

A.     Change  of  state  of  aggregation. 

130.  THE  questions  which  we  now  proceed  to  discuss 
were  first  examined  by  elementary  methods.  Some  of  the 
results  have  been  already  given.  The  rest  are  chiefly  due 
to  M.  Moutier.  The  thermodynamic  potential  was  first 
applied  to  the  questions  by  Duhem  in  1886,  and  they  will 
be  found  to  afford  very  simple  illustrations  of  its  use. 

We  suppose  that  two  portions  of  the  substance  can 
coexist  in  stable  equilibrium  in  two  different  states  at  the 
same  temperature  and  pressure,  and  that,  consequently, 
it  is  possible  to  pass  from  one  state  to  the  other  by  a 
reversible  operation  during  which  the  temperature  and 
pressure  remain  constant.  There  will  also  be  states  of 
unstable  equilibrium  which  may  be  obtained  thus.  Be- 
ginning with  a  state  of  stable  equilibrium  and  keeping 
the  pressure  constant,  we  endeavour  to  raise  or  lower  the 
temperature ;  or,  keeping  the  temperature  constant,  we 
endeavour  to  increase  or  diminish  the  pressure,  without 


APPLICATIONS   OF   THE  THERMODYNAMIC   POTENTIAL.     345 

causing  the  substance  to  pass  from  one  state  to  the  other. 
We  then  wish  to  be  able  to  distinguish  between  stable 
and  unstable  states  of  equilibrium  when  the  system  is 
contained  in  a  vessel  of  any  kind  and  subjected  to  no 
external  forces  but  a  uniform  and  constant  normal  surface 
pressure.  For  this  purpose  it  seems  natural  to  employ 
the  thermodynamic  potential  at  constant  pressure.  The 
fact  that  we  are  unable  to  take  6  and  p  as  independent 
variables  to  define  the  state  of  the  system  makes  no 
difference  to  the  two  principal  properties  of  the  thermo- 
dynamic potential  —  that  its  variation  is  always  negative 
and  that,  when  it  becomes  a  small  quantity  of  the  second 
order,  the  equilibrium  is  stable. 

Let  the  system  to  be  considered  consist  of  a  mass  w  in 
the  state  (w)  and  a  mass  s  in  the  state  (s),  at  the  same 
temperature  0  and  pressure  p.  Also  let  the  thermo- 
dynamic potential  at  constant  pressure  of  unit  mass  of  the 
substance  in  the  first  state  be  <&w  and  in  the  second  state 
3>s  :  <f>w  and  <&s  both  being  functions  of  p  and  6.  Then  if 
<I>  be  the  thermodynamic  potential  at  constant  pressure  of 
the  whole  system,  we  shall  have 


If  a  mass  ds  pass  from  the  state  (w}  to  the  state  (s)  by 
any  path  whatever  in  which  the  external  pressure  is 
constantly  equal  to  p  and  if  the  temperature  be  the  same 
at  the  end  of  the  process  as  at  the  beginning,  we  have, 
since  <&w  and  3>8  are  unaltered, 

d3>  =  (3?s-3>w)ds. 

Now  d<&  must  be  negative.  Hence  if  <E>g  be  greater  than 
3>w,  the  system  will  be  unable  to  pass  from  the  state  (w) 
to  the  state  (s),  but  it  may  pass  from  (s)  to  (w).  If  <i>g  be 


346  ELEMENTARY  THERMODYNAMICS. 

less  than  $>w,  the  system  will  only  be  able  to  pass  from 
the  state  (w)  to  the  state  (s).  If  <£s  be  equal  to  3>w,  the 
equilibrium  will  be  stable,  the  system  being  unable  to 
pass  either  from  (w)  to  (s)  or  from  (s)  to  (w). 

From  the  equation  <&S  =  <&W)  we  see  that  p  will  be  a 
function  of  6  when  the  two  states  of  the  substance  are  in 
stable  equilibrium  together  at  the  same  temperature  and 
pressure.  This  conclusion  has  already  been  drawn  in  the 
case  of  the  solid  and  liquid  states,  and  might  have  been 
proved  in  the  same  way  in  other  cases. 

If  (p  +  dp,  6  +  dO)  be  consecutive  values  of  (p,  6} 
satisfying  the  equation  <X>g  =  <&w,  we  obtain 

d<l></7«  0.^/70     d®u'j    <d®«>ja 
-y—  dp  +  -y  -  act  =  —  —  dp  H  —  T.S-  dO. 

dp  dd  dp  dO 

But  we  may  take  6  and  p  as  independent  variables  when 
the  substance  is  entirely  in  one  state.  Hence  if  the 
volume  and  entropy  of  unit  mass  in  the  two  states  be 
denoted  by  the  symbols  (vw,  tf>w)  and  (vs,  <£s),  respectively, 
we  get 

vsdp  —  (f>sdfl  =  vwdp  — 


and  therefore  <j>8  -  </>w  =  (vs  -  vw)  -?  , 

~P  being  found  from  the  equation  <&s  =  <J>W. 

If  L  be  the  latent  heat  of  transition  from  the  state  (w) 
to  the  state  (s),  we  have 


L 
0 

and  thence 


(134). 


APPLICATIONS  OF  THE   THERMODYNAMIC  POTENTIAL.     347 


Also  if  we  take  two  rectangular  axes,  and  measure 
temperature  along  Ox  and  pressure  along  Oy,  the  equation 
4>8  =  <PW  will  represent  a  curve  on  opposite  sides  of  which 
<£*  -  QIC  will  have  different  signs. 

For  if  a  straight  line  be  drawn  parallel  to  06,  cutting  the 
curve   in   a   point   P  whose   coordinates   are   (0,  p),  the 
P  / 


/P       Q' 


coordinates  of  a  point  Q  a  little  to  the  left  may  be  written 
(0-d0,p),  and 

&.(0  -  dd,  p)  -  &w(0  -  de,  p)=  $.  -  d0  ^  - 

>s     d®w\ 

de  —  af) 

=  de  (&-<}>„) 


~T  ' 

The  coordinates  of  a  point  a  little  to  the  right  will  be 
(0  +  d0,p),  and 


>.  (0  +  de,  p)  -  &„  (6  +  d0,p)  =  3>s 


d0L 

e  ' 


348  ELEMENTARY   THERMODYNAMICS. 

The  equation  d<&  =  (<3>s  —  <E> 


in  the  first  case,  and 


ds  then  becomes 

d0Lds 
0 

dOLds 


in  the  second. 

Hence  we  conclude  that  the  only  event  possible  on 
the  right  of  the  curve  is  that  which  absorbs  heat ;  and  on 
the  left  of  the  curve,  that  by  which  heat  is  evolved.  For 
example,  if  a  vessel  contain  water  and  saturated  steam, 
and  we  can  manage  to  raise  the  temperature  without 
altering  the  external  pressure  or  causing  the  water  to 
evaporate,  the  only  phenomena  possible  will  be  evapo- 
ration of  the  water  and  consequent  absorption  of  heat. 
But  if  the  temperature  be  reduced,  the  only  phenomena 
possible  will  be  condensation  of  steam  and  evolution  of 
heat. 

Again,  if  a  line  be  drawn  through  P  parallel  to  OP, 
the  coordinates  of  a  point,  R,  a  little  above  P  may  be 
written  (0,  p  +  ap),  and 

^s (0,p  +  dp)  -  <E>W  (0, p  +  dp)  =  <&s  +  dp  -~  —<&w  —  dp—r^ 


^  P 

R-- 


APPLICATIONS   OF   THE   THERMODYNAMIC   POTENTIAL.      349 

The  coordinates  of  a  point,  R',  a  little  below  P  will  be 
(0,  p  —  dp),  and 

$>s  (0,P  -  dp)  -  ®w  (d,P  -  dp)  =  d>s-  dp    '  -  ow  +  dp^y 


=  -(vs-  vw)  dp. 

Hence  at  all  points  above  the  curve,  the  only  event 
possible  is  a  change  to  the  state  of  smaller  volume  ;  below 
the  curve,  to  the  state  of  greater  volume.  Thus  if  a 
vessel  contain  water  and  saturated  steam,  and  we  succeed 
in  increasing  the  pressure  without  altering  the  tempe- 
rature or  causing  the  steam  to  condense,  the  water  will  be 
unable  to  evaporate  but  the  steam  will  be  in  danger  of 
liquefying.  If  we  reduce  the  pressure  without  altering 
the  temperature  or  causing  the  water  to  evaporate,  the 
steam  will  be  unable  to  condense  and  the  water  will  be  in 
danger  of  explosive  evaporation. 

131.  If  the  substance  which  can  exist  in  two  different 
states  is  contained  in  a  vessel  of  given  volume,  we  must 
use  the  thermodynamic  potential  at  constant  volume. 

The  thermodynamic  potential  at  constant  volume  of 
unit  mass  in  the  state  (w)  will  be  a  function  of  6  and  vw 
which  may  be  denoted  by  fw  (6,  vw).  In  like  manner,  the 
thermodynamic  potential  at  constant  volume  of  unit  mass 
in  the  state  (s),  at  the  same  temperature  0,  may  be  written 
^s  (Q,  vs)-  Thus,  if  f  be  the  thermodynamic  potential  at 
constant  volume  of  the  whole  system, 

7  =  wfw  (0,  vw)  +  sFs  (d,  vg). 


Hence,  if  a  mass  ds  pass  from  the  state  (w)  to  the 
state  (s)  by  any  path  whatever  during  which  the  volume 


350  ELEMENTARY  THERMODYNAMICS. 

is  constant,  and  if  the  temperature  be  the  same  at  the 
end  of  the  process  as  at  the  beginning,  we  shall  have 


Now  in  a  state  of  stable  equilibrium,  we  have  c£F  = 
also^s=^>  .     Thug 

dvs        dvw 

(F8  -  f^  ds-p  (sdvs  +  wdvw)  =  0. 
But  since  svg  +  wvw  and  s  +  w  are  both  constant,  we  have 

sdvs  +  wdvw  =  —  (v8  —  vw)  ds. 
Hence  in  a  state  of  stable  equilibrium, 


or,  since  3>s  =  f  s  +  pvs  and  <E>W  =  f  w  +  pvw  , 

«V=$W, 

which  is  the  very  same  equation  as  was  found  to  hold 
when  the  external  pressure  was  constant.  The  conditions 
and  properties  of  stable  equilibrium  are  therefore  the 
same  whether  the  pressure  or  the  volume  be  invariable. 

132.  If  the  substance  can  exist  in  a  third  state  (i), 
the  condition  that  the  state  (s)  may  be  in  stable  equili- 
brium with  the  state  (i)  is  <&s  =  ®t.  Hence  when  the 
same  values  (pr,  6')  of  (p,  6)  satisfy  the  two  equations 
^g  =  *&w  and  ^>g  =  <J)i,  we  have 


From  these  two  equations,  we  obtain 


APPLICATIONS   OF  THE   THERMODYNAMIC   POTENTIAL.     351 

We  therefore  see  that  the  two  states  (w)  and  (i)  can  also 
exist  permanently  together  at  the  pressure  p  and  tempe- 
rature &  ',  which  thus  constitutes  a  triple  point. 

If  we  take  two  rectangular  axes  to  denote  temperature 
and  pressure,  we  shall  have  three  curves  corresponding  to 
the  three  equations 


These  curves  are  such  that  if  any  point  be  common  to 
two  of  them,  the  third  will  also  pass  through  the  same 
point.  But  we  know  that,  in  general,  there  is  but  one 
such  point.  The  relative  position  of  the  three  curves  will 
therefore  be  known  if  we  are  able  to  determine  their 
distribution  in  the  neighbourhood  of  this,  the  triple,  point. 
At  any  given  temperature  0,  let  us  suppose  that  the 
equations  to  the  three  curves  give  the  values  (p1}  pz,  ps), 
respectively,  for  p.  Then  when  6  is  very  near  to  6',  we 
obtain 


Hence 


352 


ELEMENTARY  THERMODYNAMICS. 


If  the  volumes  of  unit  mass  in  the  three  states  be  denoted 
at  the  triple  point  by  (vsf,  vw',  V{),  respectively,  we  get,  by 
rejecting  small  quantities  of  the  second  order, 

(Pa  ~  Pi)  (v*   ~  Vi)  =  (Ps  ~  Pi)  (Vw  ~  Vt). 

Similarly, 

(Pa  ~ PS) (Vw  ~ Vi)  =  (PJ.  - p.2) (vw' - Vg), 

(Pi  ~  Pz)  (Vg  ~  Vw')  =  (p2  -pa)  (Vg  -  Vi). 

In  the  case  of  water  in  the  three  forms  of  steam,  water 
and  ice,  the  volumes  arranged  in  order  of  magnitude  are 
known  to  be  (vs,  V{,  vw). 
Also  at  a  temperature  a  little  below  6',  the  greatest  pres- 


Steam  line. 


O  <9-273 

sure  is  known  to  be  p3.     The  three  pressures  arranged  in 
order  of  magnitude  are  therefore  (p2,  plip3). 

More  generally,  if  a  line  be  drawn  parallel  to  Op, 
intersecting  the  curves  in  three  points,  the  middle  point 
will  belong  to  that  curve  which  corresponds  to  the  greatest 
change  of  volume. 


APPLICATIONS   OF  THE   THERMODYNAMIC   POTENTIAL.     353 


B.     Saline  Solutions. 

133.  The  vapour  emitted  by  dilute  sulphuric  acid,  or 
an  aqueous  solution  of  a  salt,  is  known  to  be  pure  aqueous 
vapour.  We  also  know  that  a  solution  can  be  in  equi- 
librium with  the  salt  but  not  with  water  so  that  there 
is  a  maximum  but  not  a  minimum  limit  to  the  strength 
of  the  solution.  We  now  propose  to  examine  some  of  the 
properties  of  solutions.  The  methods  of  the  thermo- 
dynamic  potential  were  first  employed  for  this  purpose  by 
Helmholtz  in  some  memoirs  on  electrical  subjects  which 
have  been  translated  by  the  Physical  Society.  A  compre- 
hensive and  simple  investigation  has  also  appeared  in 
Duhem's  work  on  the  '  Thermodynamic  Potential '. 

The  systems  to  be  considered  are  supposed  to  be  sub- 
jected to  no  external  forces  but  a  uniform  and  constant 
normal  surface  pressure,  and  we  therefore  make  use  of 
the  thermodynamic  potential  at  constant  pressure. 

Let  a  homogeneous  solution  in  a  state  of  equilibrium 
be  composed  of  a  mass  s  of  salt  and  a  mass  w  of  water. 
Also  let  the  constant  surface  pressure  be  p  and  the  uni- 
form temperature  of  the  solution  9.  Then  <£sw,  the 
thermodynamic  potential  at  constant  pressure  of  the 
solution  will  be  a  function  of  the  four  variables 
(6,  p,  s,  w). 

Now  if  X  be  any  positive  quantity,  and  we  take  a 
second  homogeneous  solution  at  the  same  pressure  and 
temperature  and  of  the  same  composition  as  the  first  but 
X  times  as  great,  its  thermodynamic  potential  at  constant 
pressure  will  evidently  be  X^^.  Thus  when  we  multiply 
P.  23 


354  ELEMENTARY  THERMODYNAMICS. 

both  s  and  w  by  the  same  factor  X  without  altering  6 
or  p,  we  also  multiply  <&sw  by  X.  We  therefore  conclude 
that  as  far  as  the  variables  s  and  w  are  concerned,  Osw 
is  a  homogeneous  function  of  the  first  degree.  Hence  if 
we  put 


—  7          -s>      —  7  —      -w> 
ds  dw 

we  obtain,  by  Euler's  theorem  of  homogeneous  functions, 
3>sw  =  sFs  +  wFw  ...............  (135). 

The  functions  Fs  and  Fw  being  evidently  homogeneous 
and  of  the  degree  0  in  s  and  w,  we  have  again  by  Euler's 
theorem, 

dF         dFs 

S-Y-  +  W  ,—  = 

ds          dw 


^  = 

ds  dw         ) 

or  since  dF*     dF™     **•» 

01  ,  biiioe  —  —  =  —  -  —  =  —  —  -  — 

dw       ds       dsdw 


............  (136). 

s        at  w 
-j  —  \-w^-^-  =  0 
dw         dw 


134.  If  a  system  be  composed  of  the  homogeneous 
saline  solution  just  considered  together  with  a  mass  fis 
of  free  salt  at  the  bottom  of  the  solution  and  a  mass  fj,w 
of  water-vapour  above  it,  in  equilibrium  at  the  pressure  p 
and  temperature  0,  the  thermodynamic  potential  at  con- 
stant pressure  of  the  system  will  be 


Q  =  sFl  +  wF 
where  %  and  Ww  are  the  thermodynamic  potentials  at 


APPLICATIONS   OF   THE   THERMODYXAMIC   POTENTIAL.      355 

constant  pressure  of  unit  masses  of  salt  and  water-vapour, 
respectively,  at  the  pressure  p  and  temperature  6. 

If  a  small  change  of  state  occur  in  consequence  of 
which  s  increases  by  ds,  /*g  will  at  the  same  time  decrease 
by  ds.  Thus  if  0,  p  and  w  are  not  affected  by  the  change 
of  state,  we  have,  by  equation  (110), 


Hence,  when  the  solution  is  in  stable  equilibrium  with 
salt,  in  which  case  it  is  said  to  be  '  saturated  ', 

Fs  =  *s  ...............  ......  (137). 

If  we  write  h  for  —  ,  the  '  strength  '  of  the  solution,  this 
w 

result  becomes  a  relation  between  (6,  p,  h)  which  shows 
that  the  strength  of  a  saturated  solution  depends  only  on 
the  temperature  and  pressure. 

In  like  manner,  when  the  solution  is  in  equilibrium 
with  aqueous  vapour,  we  find 

FW  =  V«  .....................  (138), 

from  which  we  conclude  that  the  pressure  of  the  aqueous 
vapour  which  is  in  stable  equilibrium  with  a  solution 
depends  only  on  the  temperature  and  on  the  strength  of 
the  solution. 

135.  Now  suppose  that,  at  a  given  pressure  p  and 
temperature  0,  we  have  two  solutions,  the  first  of  which  is 
formed  by  dissolving  a  mass  s  +  ds  of  salt  in  a  mass  w  of 
water,  and  the  second  by  dissolving  a  mass  s  —  ds  of  salt 
in  another  mass  w  of  water:  also  suppose  the  pressure 
and  temperature  to  be  such  that  the  solutions  can  give 

23—2 


356  ELEMENTARY   THERMODYNAMICS. 

up  neither  salt  nor  steam.  Then,  on  mixing  the  two 
solutions  together,  we  get  a  new  solution  composed  of  a 
mass  2s  of  salt  dissolved  in  a  mass  2w  of  water. 

The  thermodynamic  potential  at  constant  pressure  of 
the  two  solutions,  before  being  mixed,  is 

0  (6,  p,  s  +  ds,  w)  +  3>  (0,  p,  s  —  ds,  w), 

and  after  mixing,  if  the  temperature  and  pressure  be  the 
same  as  before, 

O  (0,  p,  2s,  w),  or  2<£  (6,  p,  s,  w). 

The  operation  being  irreversible,  the  thermodynamic  po- 
tential will  have  decreased.  Hence 

2<I>  (6,  p,  s,w}<3>  (6,  p,  s  +  ds,  w)  +  <£  (6,  p,  s  -  ds,  w), 


If  we  remember  that  ~^  =  fi,  this  inequality  shows  that 

/7  7F 

~  is  positive.     But  since  Ft  is  a  function  of  (6,  p,  h) 

where  h  =  -1,  we  have  ^  =  1  ^  .    Hence  ^  is  positive 
w  ds      w  dh  dh       ^ 

and  Fs  continually  increases  with  the  strength   of  the 
solution. 

In  like  manner  we  may  prove  that  -—-  is  positive; 
and    smce  -^  =  --i  ^,   we    conclude   that  Fw  con- 


APPLICATIONS   OF   THE   THERMODYNAMIC   POTENTIAL.      357 

tinually  decreases  as  the  strength  of  the  solution  increases. 
This  result  may  also  be  deduced  from  the  former,  for,  by 
equation  (136),  we  have 

^  +  <f»  =  0, 
ds       ds 

11       f       j  dFs     dFw     n      -T-T         •  r  dFg  ,  .  . 

and  therefore  h  —rr-  H  —  ^-  =  0.     Hence  if  -77-  be  positive, 
dh       dh  dh 

—j~  must  be  negative. 

If  we  take  two  rectangular  axes  to  denote  temperature 
and  pressure,  equation  (138)  will  give  a  series  of  curves 
for  different  values  of  h  representing  the  relation  between 
the  vapour-pressure  and  the  temperature  and  no  two  of 
these  curves  will  intersect.  For,  if  possible,  let  the  curves 
belonging  to  two  different  values  of  h  intersect  in  the 
point  (6,  p).  Then,  by  equation  (138)  we  have 


so  that  Fw  (6,  p,  h)  =  Fw  (0,  p,  h'). 

But  Fw  continually  decreases  as  h  increases  and  therefore 
cannot  have  the  same  value  for  two  different  values 
of  h. 

Hence  if  a  curve  be  drawn  to  express  the  relation  between 
the  pressure  and  temperature  of  saturated  steam,  which 
will  coincide  with  the  curve  Fw  (0,  p,  h)  =  tyw  (6,  p)  when 
h  is  zero,  the  whole  series  of  curves  will  lie  entirely  on 
one  side  of  it,  further  and  further  away  as  h  increases 
from  zero.  To  determine  their  relative  positions  it  will 
therefore  be  sufficient  to  know  the  position  of  the  curve 
for  which  h  is  very  small  with  respect  to  the  curve  for 
which  h  is  zero.  To  find  this,  we  may  proceed  as  follows. 


358  ELEMENTARY  THERMODYNAMICS. 

Differentiating  equation  (138)  on  the  supposition  that  6 
is  constant,  we  get 

~dh+~dp  ~dh  =   dp     dh  ' 

But  if  vw  be  the  volume  of  unit  mass  of  aqueous  vapour 
in  stable  equilibrium  with  the  solution,  we  have 


Hence  ^  =  k~^F   If (139). 


Now  in  obtaining  the  formula  which  gives  the  volume  in 
a  state  of  stable  equilibrium  in  terms  of  the  thermo- 
dynamic  potential  at  constant  pressure  in  the  form 

vdp  =  ®(p  +  dp)  —  <E>  (p), 

we  made  no  supposition  except  that  the  temperature 
remains  constant  and  that  a  small  change  of  p  occasions 
only  a  small  change  of  state.  Thus,  if  vg  be  the  volume 
of  unit  mass  of  the  solution,  we  have 

(s  +  w)  vsdp  =  d  (sFs  +  wFw), 

where,  on  the  right  hand  side,  we  may  either  take  h 
constant  or  variable.  If  we  wish  to  take  h  variable,  let  a 
small  arbitrary  quantity  of  vapour,  dfjuw,  be  formed,  in 
consequence  of  which  w  becomes  w  —  dpw.  Then 

d  (sFg  +  wFw)  =  A  (SFS  +  wFw)  dp 


APPLICATIONS   OF  THE   THERMODYNAMIC   POTENTIAL.      359 


by  equations  (136)  and  (138).     Therefore 

dFs        dFw 
=  s—  +  W-^, 


7TT 

Thus  as  h  tends  to  zero,   -~  will  approximate  to  the 

volume  of  unit  mass  of  water.     This  being  less  than  vw, 

dF 
we  see  from  equation  (139),  that  when  h  is  small,  -^- 

and  -^-  are  of  the  same  sign,  so  that  -~-  is  negative. 
Hence  the  curves  all  lie  below  the  curve  for  which  h  =  0, 


and  therefore  at  any  given  temperature  the  vapour- 
pressure  continually  decreases  as  the  strength  of  the 
solution  increases. 

136.  If  a  solution  contained  in  a  vessel  in  a  state  of 
stable  equilibrium  be  subjected  to  such  a  pressure  that 
the  vapour  is  just  on  the  point  of  forming,  and  if,  by 


360  ELEMENTARY  THERMODYNAMICS. 

slowly  diminishing  the  pressure,  a  small  amount  of  vapour 
d/j,w,  be  formed  in  a  reversible  manner,  a  small  quantity 
of  heat  which  may  be  written  Ldpw,  will  be  required  to 
keep  the  temperature  constant  during  the  process.  This 
quantity  satisfies  the  equation 


where  the  suffixes  refer  to  the  two  states  of  the  system. 
But  when  the  pressure  is  kept  constant  and  a  small  change 
of  temperature  can  only  produce  a  small  change  of  state, 
we  have,  when  the  pressure  remains  constant, 


If,  for  convenience,  we  suppose  6  to  be  the  only  quantity 
that  varies,  this  result  takes  the  simple  form  —  </>  =  ja  . 

Ldfjiw       dFs  (6,  p,  5,  w)        dFw  (d,  p,  s,  w) 
Hence         — -~ •  =  s  —   ^jfi      —  +  w  —     v     •%  — 

m  dFs  (6,  p,s,w-  dp*,)  ,    .dFw(e)p,s,w-dfjiw) 

~s~      ~dd~       -(»-«W-      -^— 


and  therefore 

L      d   (  dFs  (0,  p,  s,  w)        dFw  (0  p  s  w) 

0=d0\s dw +  ^-—^-+Fw(0,p,s,w) 


=  j0(Fw~  ^'  b^  ecluation  (136). 
Also  by  differentiating  the  equation  Fw  =  Vw,  we  get 

-(F  -V}-     d(F™-^dhP 
de(J*w     ^w)_        __ __. 


APPLICATIONS  OF  THE  THERMODYNAMIC   POTENTIAL.      361 


Thus  finally  L  =  6   vv-  ............  (140). 

\         dp  J  dv 

Since  —^  will  be  small  in  comparison  with  vw,  we  obtain 
approximately 


137.     Kirchhoff's  formula  may  be  easily  deduced  by 
means  of  the  thermodynamic  potential.     For  when  the 
pressure  is  constant,  the  equation 
dU=dQ-pdv 
becomes  dQ  =  d(U+pv)  ...............  (141) 

Now  we  have  <E>  =  U  —  6$  +  pv, 

and  in  a  state  of  stable  equilibrium 


Hence  <&-  6^^  =  U+  pv, 

da 

and  equation  (141)  becomes 

(142). 


If  the  operation  consist  of  adding  a  quantity  d/j,w  of  water 
to  a  solution  formed  of  a  mass  s  of  salt  dissolved  in  a  mass 
w  of  water,  and  if  ^  (0,  p)  be  the  thermodynamic  potential 
at  constant  pressure  of  unit  mass  of  water, 


Therefore 


362  ELEMENTARY  THERMODYNAMICS. 


and  thence  - 

dh  \dfjij       dh 

Now  since  a  liquid  is  very  little  affected  by  pressure,  the 

value  of  —fr  will  be  practically  independent  of  p.     But 
ct/i 

when  the  pressure  is  equal  to  that  of  the  vapour  of  the 
solution,  we  have  approximately,  by  equation  (139), 

dF_w  =  v   dep 
dh        w  dh-' 

and  the  vapour  is  generally  so  near  the  state  of  the  ideal 

perfect  gas  that  we  shall  introduce  no  serious  error  by 

putting  pvw  =  RO,  where  R  is  a  constant.     Hence 

dFw=R0dep 

dh        p    dh 


and  consequently 


dhdd    ' 

If  the  pressure  of  saturated  steam  at  the  temperature 
6  be  denoted  by  P,  Kirchhoffs  formula  for  the  heat 
absorbed  is  obtained  on  integration  :  thus 


138.  The  principles  of  thermodynamics  were  first 
applied  to  the  freezing  of  saline  solutions  in  1886  by 
Duhem  in  his  work  on  the  Thermodynamic  Potential. 
His  investigations  may  be  easily  completed  by  aid  of 
Dr  Guthrie's  beautiful  experimental  researches  which 
may  be  seen  in  the  Philosophical  Magazine,  1875—1876. 


APPLICATIONS  OF   THE  THEKMODYNAMIC   POTENTIAL.      363 

When  an  aqueous  solution  of  a  salt  is  slowly  reduced 
in  temperature,  it  is  supposed  that  the  ice  which  freezes 
out  is  perfectly  pure.  This  appears  to  be  strictly  estab- 
lished by  experiment  when  the  solution  contains  no  floating 
impurities  and  the  cooling  is  so  slow  that  the  liquid  solu- 
tion does  not  become  entangled  among  the  ice. 

Let  a  system  in  a  state  of  equilibrium  be  composed  of 
a  homogeneous  solution  formed  of  a  mass  s  of  salt  dissolved 
in  a  mass  w  of  water,  together  with  a  mass  /^  of  pure  ice 
and  a  mass  ps  of  free  salt,  not  in  contact  with  each  other 
or  with  the  solution.  Also  let  the  thermodynamic  potential 
at  constant  pressure  of  unit  mass  of  pure  ice  be  M^  (0,  p). 
Then  the  thermodynamic  potential  at  constant  pressure  of 
the  whole  system,  at  the  same  temperature  9  and  pressure 

P,  is 

3>  =  sFg  +  wFw  +  ftp,  +  fi&i. 

If  a  small  quantity  of  ice  fall  into  the  solution  in 
consequence  of  which  w  increases  to  w  +  dw,  fn  will  at 
the  same  time  decrease  by  dw.  The  temperature  being 
brought  to  its  original  value  and  s  remaining  constant,  it 
is  easily  shown  that 


This  is  always  negative.     Hence  the  only  phenomenon 
possible  will  be  the  formation  of  ice  when  Fw  —  ^  is 
positive,  and  the  disappearance  of  ice  when  Fw  —  ^i  is 
negative.    The  condition  that  the  system  may  be  in  stable 
equilibrium  when  the  ice  is  in  contact  with  the  solution  is 
Fw(d,p,h)-Vi(e,p)  =  V  .........  (144), 

an  equation  which  admits  but  one  real  value  of  h  when  Q 
and  p  are  given,  since  when  6  and  p  are  constant,  Fw 
continually  decreases  as  h  increases. 


364  ELEMENTAEY  THERMODYNAMICS. 

In  like  manner,  if  s  increase  to  s  +  ds,  and  (0,  p,  w)  be 
unchanged,  we  have 


and  therefore  the  condition  that  the  solution  may  be  in 
stable  equilibrium  when  the  solution  and  free  salt  are  in 
contact,  is 

^-^  =  0, 

which  is  easily  seen  to  have  but  one  real  root  in  h. 

Again,  it  is  well  known  that  at  ordinary  temperatures, 
when  salt  and  ice  or  salt  and  snow  are  mixed  together,  a 
violent  chemical  action  ensues  by  which  the  temperature 
is  greatly  reduced  and  a  liquid  solution  formed.  The 
thermodynamic  potential  at  constant  pressure  of  a  solution 
will  therefore  be  less  than  the  sum  of  the  thermodynamic 
potentials  of  the  salt  and  ice  from  which  it  is  formed. 
Thus 


Hence,  when  FS-VS  =  Q,  Fw-^t  is  negative.  As  h 
decreases  from  this  value,  Fs  —  M>s  becomes  negative  and 
Fw  —  M^i  continually  increases.  It  therefore  follows  that 
the  value  of  h  which  makes  Fw  —  ¥t-  =  0  is  less  than  that 
which  makes  F8  -  %  =  0.  Also  for  any  value  of  h  between 
these  two  limits,  F8  -  M*,  and  Fw  —  ^  are  both  negative. 
The  solution  is  then  capable  of  dissolving  either  salt  or 
ice,  and  is  stable  if  neither  salt  nor  ice  be  present. 

139.  If  we  take  two  rectangular  axes  of  9  and  p, 
equation  (144)  will  give  a  series  of  curves  for  different 
values  of  h  which  show  how  the  pressure  must  vary  with 
the  temperature  when  the  solution  is  in  stable  equilibrium 
with  pure  ice  to  keep  the  concentration  of  the  solution 


APPLICATIONS   OF   THE   THERMODYNAMIC   POTENTIAL.      365 

constant.  No  two  of  these  curves  can  intersect,  for  if  the 
curves  belonging  to  two  different  values  of  h  could  pass 
through  the  same  point  (6,  p),  we  should  have 


which  is  impossible,  since  Fw  continually  decreases  as  h 
increases. 

The  curves  will  therefore  all  lie  on  one  side  of  that  for 
which  h  is  zero.  To  find  their  relative  dispositions,  we 
differentiate  equation  (144)  on  the  supposition  that  0  is 
constant:  thus 

dFw      fdFw     d&i\  dep 

—  57—  +       ^7  ---  7  -       —  VT   =  "• 

dh       \dp       dp  J  dh 
But  if  Vi  be  the  volume  of  unit  mass  of  ice,  we  have 


Hence 

p 

Also  when  we  restrict  ourselves  to  a  curve  for  which  h  is 
indefinitely  small,  the  equation 

,  dF8     dFw 

(h  +  l)vg  =  h-^  +  -—- 
dp       dp 

7  77T 

shows  that  —^  is  equal  to  the  volume  of  unit  mass  of  the 
dp 

solution,  which  is  the  same  as  vw,  the  volume  of  unit  mass 
of  water.     Thus 


The  factor  Vi  —  vw  being  positive,  because  water  expands 
in  freezing,  we  see  that  when  h  is  small,  -      is  negative. 


366 


ELEMENTARY  THERMODYNAMICS. 


All  the  curves  therefore  lie  below  the  curve  of  fusion  of 
pure  ice,  that  is,  below  the  'ice  line.'     Hence  when  the 

P 


pressure  is  given,  the  temperature  at  which  ice  begins  to 
separate  out  of  a  solution  is  lower  for  a  strong  than  for 
a  weaker  solution. 

When  a  salt  is  dissolved  in  a  liquid,  like  acetic  acid, 

which  contracts  in  freezing,  v{  —  vw  is  negative,  -^  is  also 

negative,  as  before,  and  therefore  when  h  =  Q,  ~-  is  positive. 

In  this  case,  the  curves  corresponding  to  different  values 
of  h  all  lie  above  the  curve  for  which  h  =  0.  But  on  this 


o. 


APPLICATIONS   OF   THE   THERMODYNAMIC   POTENTIAL.      367 


curve,  unlike  the  curve  for  water,  0  and  p  increase  together. 
Hence,  as  before,  when  the  pressure  is  given,  the  tempera- 
ture at  which  the  pure  frozen  liquid  begins  to  appear,  is 
lower  for  a  strong  than  for  a  weaker  solution. 

Thus  in  all  cases,  the  presence  of  a  salt  retards  the 

formation  of  ice  from  a  liquid  by  cold,  or  -—•  is  always 
negative  for  the  stable  solution  of  minimum  strength. 

140.  If  we  suppose  that  every  liquid  can  be  reduced 
to  the  solid  state  by  applying  a  sufficient  degree  of  cold, 
it  will  follow  that  salt  and  ice  are  neutral  to  one  another 
when  the  temperature  is  low  enough.  If  a  solution  can 
then  exist,  it  must  be  in  stable  equilibrium  with  both  salt 
and  ice  simultaneously.  The  temperature  at  which  this 
FIRST  takes  place  and  the  strength  of  the  corresponding 
solution,  are  given  by  the  equations 


If  we  take  two  rectangular  axes  to  represent  9  and  h, 
k 


these  equations  will  give  two  curves  for  each  value  of  p, 
intersecting  in  a  point  C  whose  coordinates  represent  the 


368  ELEMENTARY  THERMODYNAMICS. 

temperature  at  which  ice  and  salt  first  become  neutral  for 
that  value  of  p,  and  the  strength  of  the  only  stable  solution 
which  can  then  exist. 

Let  us  take  a  mixture  of  ice  and  salt  at  the  highest 
temperature  at  which  they  are  neutral  to  one  another, 
and  then,  by  the  application  of  heat,  let  a  liquid  solution 
be  formed  without  altering  either  the  temperature  or  the 
pressure.  The  operation,  though  an  equilibrium  operation, 
will  be  irreversible,  for  it  is  found  that  if  we  attempt  to 
reduce  the  temperature,  we  do  not  dissociate  the  solution 
into  its  original  salt  and  ice — what  really  occurs  will  be 
seen  presently.  If  the  solution  be  formed  of  a  mass  s  of 
salt  dissolved  in  a  mass  w  of  water,  its  thermodynamic 
potential  at  constant  pressure,  <I>,  will  be 
3>  =  sFg  +  wFw, 

or,  since  *  ~~ 


The  thermodynamic  potential  is  therefore  unaltered  by 
the  process.  But  in  any  finite  change  of  state  in  which 
the  temperature  and  pressure  remain  constant,  we  have 


The  equation  J   -^  =  $B  -  <j>A  is  thus  proved  to  be  true  for 
this  particular  irreversible  process. 

141.  It  will  now  be  necessary  to  introduce  some 
additional  experimental  results,  all  of  which  are  due  to 
Dr  Guthrie.  They  have  only  yet  been  shown  to  hold  at 
atmospheric  pressure,  but  we  shall  suppose  them  to  be 
true  at  any  pressure. 


APPLICATIONS   OF   THE   THERMODYNAMIC   POTENTIAL.      369 

It  is  found  that  if  we  take  a  stable  saline  solution  of 
any  possible  strength  and  subject  it  to  reduction  of 
temperature,  the  only  solids  which  the  solution  gives  up 
are  pure  ice  or  the  anhydrous  salt  or  a  hydrate,  until  a 
certain  definite  temperature  is  reached  which  depends 
only  on  the  nature  of  the  salt  and  liquid  which  compose 
the  solution,  and  not  on  their  relative  proportions.  The 
solution  then  freezes,  without  further  reduction  of  tem- 
perature, into  a  solid  homogeneous  brine.  The  tempera- 
ture at  which  this  takes  place  is  found  by  Dr  Guthrie  to 
be  the  same  as  that  at  which  ice  and  salt  first  become 
neutral  to  one  another.  Thence  we  infer  that  the  final 
solution  thus  obtained,  which  Dr  Guthrie  calls  a  'cryo- 
hydrate,'  is  of  the  same  strength  whatever  may  have  been 
the  strength  of  the  solution  with  which  we  started,  and 
that  the  lowest  temperature  attainable  by  means  of  a 
'  cryogen,'  that  is,  a  freezing  mixture  of  salt  and  ice,  is  the 
freezing  point  of  the  corresponding  cryohydrate. 

These  experimental  conclusions  lead  to  a  very  interest- 
ing irreversible  non-frictional  cycle  which  may  be  studied 
by  means  of  strictly  elementary  methods. 

Let  the  system  consist  originally  of  pure  ice  and  a 
salt,  such  as  nitrate  of  silver  or  chlorate  of  potassium, 
whose  solution  does  not  deposit  a  hydrate ;  or  of  ice  and 
the  hydrate  of  a  salt  like  sulphate  of  magnesium.  Also 
let  the  temperature  00  and  pressure  p0  be  such  that  00  is 
the  freezing  point  of  the  cryohydrate  at  the  pressure  p0, 
and  p0  the  pressure  of  the  saturated  vapour  of  ice  at  the 
temperature  00.  Then,  the  materials  being  mixed  together, 
the  system  may  be  made  to  undergo  the  following  cyclical 
process  at  the  constant  temperature  00  and  the  constant 
pressure  p0. 

P.  24 


370  ELEMENTARY   THERMODYNAMICS. 

(1)  By  slowly  imparting  heat,  let  a  liquid  cryohydrate 
be  formed.     Call  the  initial  state  of  system  A  and  the 
state  at  the  end  of  this  operation  B  :  also  denote  the  heat 
absorbed  in  the  operation  by  Q:. 

The  path  AB  is  evidently  irreversible,  for  by  slowly 
abstracting  heat,  we  should  not  dissociate  the  cryohydrate 
—  we  should  simply  cause  it  to  freeze. 

(2)  By  slowly  increasing  the  volume,  let  the  cryo- 
hydrate be  evaporated  at  constant  pressure  and  tempera- 
ture, giving  the  saturated  vapour  of  ice  and  the  original 
salt  or  hydrate. 

(3)  Let  the  vapour  be  separated  from  the  salt   or 
hydrate,  and  then  reduce  it  to  ice. 

The  operations  (2)  and  (3)  are  clearly  reversible  :  hence 
if  Q2  be  the  heat  absorbed  during  these  processes,  we  have 

Qi-  ft  (**-&). 

But  since  the  pressure  is  constant  throughout  the 
cycle,  the  total  quantity  of  work  done  on  the  system 
during  the  cycle,  and  therefore  also  the  total  quantity  of 
heat  absorbed,  is  zero.  Thus 

Q!  +  Q2  =  o. 

Hence,  for  the  irreversible  path  AB,  in  which  the 
temperature  is  constantly  equal  to  #0, 


142.  The  subject  we  are  studying  may  be  simplified 
by  means  of  a  diagram.  Take  three  rectangular  axes,  and 
let  the  axis  of  x  denote  the  temperature,  the  axis  of  z  the 
pressure,  and  the  axis  of  y  the  strength  of  the  solution. 
Then  the  weakest  and  strongest  stable  solutions  will  be 


APPLICATIONS  OF   THE   THERMODYNAMIC   POTENTIAL.      371 

represented  in  the  diagram  by  two  surfaces,  given  by  the 
equations 


which  meet  along  the  freezing  line  of  the  cryohydrate,  and 
all  other  stable  solutions  will  be  represented  by  points 
lying  between  these  two  surfaces. 

The  solutions  which  are  in  stable  equilibrium  with 


their  own  vapour  will  be  represented  by  a  surface  close 
to  the  plane  OOli,  whose  equation  is 


Every  point  below  this  surface  represents  a  solution  which 
is  in  danger  of  explosive  evaporation. 
The  curve  in  which  the  surface  fFw='tyw  meets  the  surface 
3F  10  =  ^1,  evidently  refers  to  those   solutions  which   are 
simultaneously  in  stable  equilibrium  with  their  own  vapour 

24—2 


372  ELEMENTARY   THERMODYNAMICS. 

and  with  pure  ice.  Its  projection,  AB,  on  the  plane  60p, 
therefore  denotes  the  relation  between  the  temperature 
and  the  pressure  of  the  saturated  vapour  of  ice :  in  other 
words,  it  is  the  '  hoar-frost  line  '  of  water. 

Again,  if  we  draw  another  curve,  AC,  in  the  plane 
60p,  to  represent  the  relation  between  the  temperature 
and  pressure  when  ice  and  water  can  exist  in  stable 
equilibrium  together,  the  surface 

fw(e,p,ii}  =  ^i(d,P) 

will  clearly  meet  the  plane  60p  in  the  curve  AC  and  will 
there  terminate,  since  h  cannot  be  negative. 

The  freezing  line  of  the  cryohydrate  may  be  called  a 
'quadruple  line,'  for  four  substances  are  in  stable  equi- 
librium with  one  another  on  it — pure  ice,  the  anhydrous 
salt,  the  liquid  cryohydrate  and  the  solid  cryohydrate.  If 
the  saturated  solution  yields  a  hydrate  on  being  cooled, 
we  must  add  a  fifth  to  the  list,  viz.,  crystals  of  the 
hydrate. 

The  freezing  line  of  the  cryohydrate  terminates  in  the 
point  in  which  it  meets  the  surface  f  'w  =  tyw.  This  point 
will  always  be  a  '  quintuple  point,'  and  when  the  solution 
gives  a  hydrate,  it  will  be  a  '  sextuple  point.' 

It  will  be  noticed  that  since  A  is  the  '  triple  point '  of 
water,  the  curve  AD  in  which  fw  =  *VW  meets  the  plane 
60p,  makes  a  finite  angle  with  the  projection  on  60p  of 
the  curve  of  intersection  of  fw  =  Vw  with  fw  =  ^ :  but 
it  should  be  remembered  that  there  is  no  discontinuity  or 
irregularity  on  the  surface  fw  =  tyw  itself. 

143.  We  have  strong  reasons  for  believing  that  h  is 
constant  along  the  freezing  line  of  the  cryohydrate.  For, 
if  possible,  let  the  strength  of  the  cryohydrate  be  greater 


APPLICATIONS   OF  THE   THERMODYNAMIC   POTENTIAL.      373 

at  a  point  Q  than  at  another  point  P  of  this  line.  Let 
the  liquid  cryohydrate  belonging  to  P  be  frozen  and  then 
travel  to  the  point  Q  by  any  path  not  lying  between  the 
surfaces 


If  the  cryohydrate  does  not  melt  before  the  point  Q  is 
reached,  it  will  then  melt  into  two  distinct  parts,  one  of 
which  is  a  liquid  cryohydrate  and  the  other  pure  solid 
ice.  If  it  could  melt  before  we  get  to  Q,  we  should  obtain 
a  solution  in  the  liquid  form  at  a  temperature  below  the 
freezing  point  of  the  cryohydrate  at  the  same  pressure. 
Both  of  these  results  are  very  improbable  and,  conse- 
quently, we  conclude  that  h  cannot  have  different  values 
at  two  different  points  P,  Q,  of  the  freezing  line  of  the 
cryohydrate. 

A  cryohydrate  must  therefore  be  regarded  as  a  body 
of  definite  chemical  composition  (as  was  foreseen  clearly  by 
Dr  Guthrie  himself),  and  not  as  a  mere  solution  in  which 
the  proportions  of  salt  and  ice  depend  on  various  acci- 
dents. 

144.     Any  two  consecutive  points  on  the  surface 

Fw(e,p,h)  =  yi(d,p) 
satisfy  the  relation 


dp 

where  the  differential  coefficients    ~  and  -~  are  to  be 

dp  d6 

found  from  the  equation  fw  (d,  p,  h)  =  ¥t-  (0,  p). 


374  ELEMENTARY  THERMODYNAMICS. 

If  the  two  points  also  lie  on  the  freezing  line  of  the 
cryohydrate,  we  have  dh  =  0,  and  therefore,  along  this 
line, 


dp 

Now  we  have  already  shown  that  when  the  salt  is 
dissolved  in  a  liquid  which  expands  in  the  act  of  freezing, 

like  water,  the  equation  fw  (6,  p,  Ji)  =  M^  (9,  p)  makes  ~ 

negative.  If,  however,  the  solution  is  formed  by  dissolving 
the  salt  in  a  liquid,  like  acetic  acid,  which  contracts  whilst 

freezing,  then  ~-  is  positive.     Also  in  both  cases,  -j^   is 

negative. 

Hence   along   the  freezing   line    of  the   cryohydrate,  ^ 

is  negative  in  the  former  case  and  positive  in  the  latter. 
Thus  the  freezing  point  of  the  cryohydrate  is  depressed  or 
raised  by  pressure  according  as  the  liquid  employed  ex- 
pands or  contracts  in  the  act  of  freezing.  In  other  words, 
when  the  liquid  employed  in  forming  the  solution  is  such 
that  its  freezing  point  is  lowered  by  pressure,  the  freezing 
point  of  the  cryohydrate  is  also  lowered  by  pressure ;  and 
when  the  liquid  employed  is  such  that  its  freezing  point 
is  raised  by  pressure,  the  freezing  point  of  the  cryohydrate 
is  also  raised  by  pressure. 

Again,  let  us  suppose  that  at  any  point  (0,  p)  on  the 
freezing  line  of  the  cryohydrate,  the  volume,  energy, 
entropy  and  thermodynamic  potential  at  constant  pressure 
of  unit  mass  of  the  cryohydrate  are  represented  in  the 
liquid  state  by  the  symbols  (vf,  U',  <£',  &\  and  in  the  solid 
state  by  («",  U",  <j>",  <£"),  respectively :  also  let  (v,  U,  <f>,  3>) 


APPLICATIONS   OF  THE   THERMODYNAMIC   POTENTIAL.     375 

be  respectively  the  sums  of  the  volumes,  energies,  entropies 
and  thermodynamic  potentials  at  the  same  temperature 
and  pressure  of  the  salt  and  ice  of  which  the  cryohydrate 
is  composed.  Then  by  Art.  140  we  have 


Also  since  the  freezing  of  a  cryohydrate  is  a  reversible 
process  performed  at  constant  temperature  and  pressure, 

<£'  =  <£". 
Thus,  at  the  point  (0,  p), 

At  a  consecutive  point  (6  +  d6,  p  +  dp),  we  have 

that  is,    vdp  —  $d6  =  v'dp  —  <j>'dO  =  v"dp  —  $"d0. 
Hence  <£'  —  </>  =  (v'  —  v)   -£ 


where  -^  refers  to  the  freezing  line  of  the  cryohydrate. 

But  if  Zj  be  the  heat  absorbed  in  the  formation  of  unit 
mass  of  the  liquid  cryohydrate  from  salt  and  ice  at  con- 
stant temperature  and  pressure,  then  L±  =  6  ($'  —  <£)  :  also 
if  L.2  be  the  heat  evolved  in  the  freezing  of  the  cryo- 
hydrate, L2  =  6  (<//  -  </>").  Thus  finally 


(145), 

A—  *<v-OJgj 

j  and  L2  both  being  positive. 


376      ,  ELEMENTARY   THERMODYNAMICS. 

An  equation  similar  to  the  first  of  these  will  hold 
when  the  liquid  solution  is  formed  by  the  melting  of  ice 
with  a  hydrate. 

Now  when  the  liquid  employed  expands  in  freezing, 

like  water,  -^  is  negative  along  the  freezing  line  of  the 

cryohydrate  :  hence,  by  equations  (145),  the  cryohydrate 
also  expands  in  freezing,  and  when  we  form  the  liquid 
cryohydrate  from  solid  ice  and  salt,  or  from  solid  ice  and 
a  hydrate,  at  constant  temperature  and  pressure,  there 
will  be  a  contraction  of  volume.  On  the  contrary,  when 
the  liquid  employed  is  one  which  shrinks  in  freezing,  the 
cryohydrate  will  also  shrink  in  freezing,  and  when  it  is 
formed  at  constant  temperature  and  pressure  from  solid 
ice  and  salt,  or  from  solid  ice  and  hydrate,  there  will  be 
an  increase  of  volume. 

145.  The  foregoing  investigations  enable  us  to  describe 
the  behaviour  of  a  saline  solution  under  various  circum- 
stances. This  will  be  made  clear  by  the  following  ex- 
amples. 

Let  the  solution  be  constantly  in  a  state  of  stable 
equilibrium:  also  suppose  that  initially  it  is  neither 
saturated  with  salt  nor  with  ice  and  that  the  pressure  is 
so  great  that  there  is  no  danger  of  evaporation. 

Then  we  may  diminish  6  without  altering  p  or  h  until 
the  right  line,  parallel  to  06,  which  represents  the  suc- 
cessive conditions  of  the  solution,  meets  one  of  the 
surfaces 


At   that   instant  a  sudden  discontinuity  occurs.     If,  for 


APPLICATIONS   OF   THE   THERMODYNAMIC   POTENTIAL.      377 

example,  the  straight  line  meets  both  surfaces  on  the 
freezing  line  of  the  cryohydrate,  it  will  still  be  possible 
to  diminish  6  without  altering  p  or  h,  but  the  solution 
will  be  frozen.  In  any  other  case  when  we  diminish  9, 
it  will  be  necessary  to  vary  one  or  both  of  the  other 
quantities  (p,  h).  Thus  if  we  wish  to  keep  p  constant,  it 
has  already  been  proved  that  any  decrease  of  6  on  the 
surface  f  ~w  =  *&i  will  make  h  increase  by  the  freezing  out 
of  pure  ice  until  the  freezing  line  of  the  cryohydrate  is 
reached  :  on  the  surface  fs  =  Ws>  it  appears  from  experi- 
ment that  when  6  is  diminished,  the  solution  generally 
deposits  the  anhydrous  salt  or  a  hydrate,  so  that  h 
decreases.  If  while  6  diminishes,  we  keep  h  constant  and 
vary  p,  we  shall  never  arrive  at  the  freezing  line  of  the 
cryohydrate,  because  along  that  line  h  has  a  different 
constant  value. 

If  we  begin  by  keeping  6  and  h  constant  and  diminish 
p,  the  successive  conditions  of  the  solution  will  be  repre- 
sented by  a  straight  line  parallel  to  Op  until  we  come  to 
one  of  the  three  surfaces 


It  will  be  sufficient  to  indicate  by  an  illustration  what 
may  happen  after  we  get  to  the  surface  fw  =  ^w.  For 
simplicity,  let  us  suppose  that  there  is  some  vapour  present 
with  the  liquid  solution. 

Then,  since  -~  has   been  shown  to  be  positive  on  the 

surface  fw  =  '^w,  we  must  diminish  h  at  the  same  time 
as  6  in  order  to  keep  p  constant.  This  may  be  done  by 


378  ELEMENTARY   THERMODYNAMICS. 

causing  the  volume  to  decrease  so  that  some  of  the  vapour 
is  condensed. 

On  the  curve  of  intersection  of  fw  =  '^w  with  f  w=^i) 
pure  ice  will  first  appear ;  and  on  the  curve  of  intersection 
with  3-g  =  ^rg,  the  solution  will  begin  to  deposit  the  salt 
or  a  hydrate  and  may  be  dissociated  by  evaporation  at 
constant  temperature  and  pressure. 


NOTE  A. 

IT  is  often  supposed  that  to  every  action  there  is 
simultaneously  an  equal  and  opposite  reaction,  whether 
these  actions  be  contact-forces  or  actions  at  a  distance. 
We  have  assumed  this  to  be  the  case  with  contact-forces, 
and  it  is  likewise  undoubtedly  true  for  the  simpler  actions 
at  a  distance,  such  as  the  gravitational  and  electro-statical 
and  magneto-statical  forces  between  bodies  at  rest  in  an 
unvarying  state.  For  example,  consider  the  mutual  gravi- 
tational influence  of  two  distant  bodies  A,  B,  which  are 
devoid  of  electric  and  magnetic  properties.  If  both  A 
and  B  be  at  rest,  it  is  quite  reasonable  to  suppose  that 
their  mutual  influence  consists  of  a  set  of  elementary 
forces  which  are  strictly  equal  and  opposite  in  pairs.  If, 
however,  while  B  is  kept  at  rest,  A  be  suddenly  moved 
nearer  or  further  away  with  great  velocity,  it  is  clear  that 
the  gravitational  force  on  A  will  at  once  begin  to  alter, 
while  that  on  B  will  remain  unchanged  until  the  effect  of 
the  motion  of  A  has  had  time  to  cross  the  space  AB — 
unless,  indeed,  we  can  conceive  the  gravitational  force 
which  acts  on  B  capable  of  foreseeing  the  intentions  of 
the  machine  by  which  the  body  A  is  moved. 

If  we  assume,  as  is  usually  done,  that  when  the  two 


380  ELEMENTARY   THERMODYNAMICS. 

bodies  have  been  held  at  rest  for  a  sufficient  time  in  an 
unvarying  state  at  a  given  distance  apart  and  in  a  given 
relative  position,  the  energy  of  the  system  of  the  two 
bodies  is  independent  of  the  previous  history  of  the 
system,  AVC  shall  be  led  to  some  important  conclusions. 
For  let  the  system  be  brought  from  any  such  condition 
(1)  to  any  other  such  condition,  (2)  in  two  different  ways, 
one  of  which  is  practically  reversible  and  the  other  very 
rapid.  Also  suppose  the  machines  by  which  the  two 
bodies  are  held  incapable  of  doing  any  but  mechanical 
work.  Then  since  the  change  of  energy  is  the  same  in 
the  two  different  methods  and  the  mechanical  work 
obtained  from  the  system  is  different,  it  follows  that 
different  thermal  processes  must  have  taken  place  in  the 
two  operations,  and  clearly  the  difference  can  only  be  due 
to  radiation. 

Again,  let  A,  B  be  two  small  distant  bodies  of  any  the 
same  uniform  temperature  0,  which  may  possess  any 
electric  and  magnetic  properties  but  cannot  gain  or  lose 
electric  energy,  and  let  them  be  in  such  an  unvarying 
state  and  position  that  the  energy  of  the  system  which 
they  compose  is  independent  of  the  previous  history  of 
the  system.  Let  this  system  be  rotated  like  a  rigid  body, 
in  a  reversible  manner,  without  varying  the  temperature, 
about  an  axis  passing  through  A.  Then  if  Q  be  the  heat 
absorbed  and  W  the  mechanical  work  done  on  the  system, 
we  have  clearly 

Q  +  W=  0, 

and  9-o 

6 

Hence  W=  0,  and  therefore  the  action  at  a  distance 
exerted  by  A  on  B  must  act  in  the  line  AB. 


NOTE  A.  381 

Similarly,  the  action  at  a  distance  exerted  by  B  on  A 
must  act  in  the  same  straight  line  AB. 
Again,  if  the  distance  AB  be  increased  or  diminished  by 
a  given  amount  in  two  isothermal  reversible  ways,  first, 
by  moving  A  along  the  line  AB  while  B  is  kept  at  rest, 
and  secondly,  by  moving  B  along  AB  while  A  is  kept  at 
rest,  and  if  we  make  the  reasonable  supposition  that  the 
changes  of  energy  and  entropy  are  the  same  in  both  ways, 
it  follows  that  the  actions  at  a  distance  between  A  and  B, 
which,  we  have  seen,  act  in  the  same  straight  line  AB, 
are  equal  and  opposite. 

If  the  two  bodies  A,  B  can  gain  or  lose  electric  energy, 
the  actions  at  a  distance  between  them  are  not  necessarily 
equal  and  opposite ;  but  since  it  is  always  immaterial 
whether  A  be  moved  in  one  direction  or  B  an  equal 
distance  with  the  same  velocity  in  an  opposite  parallel 
direction,  it  is  easily  seen  that  the  actions  at  a  distance 
are  equal  and  in  opposite  parallel  directions. 


NOTE  B. 

WE  have  already  noticed  a  point  in  which  the  nomen- 
clature of  Thermodynamics  differs  from  that  which  is 
unfortunately  adopted  in  books  on  Rigid  Dynamics.  We 
now  propose  to  consider  briefly  the  chief  problem  dis- 
cussed in  these  books. 

It  has  been  shown  that  if  a  body  which  possesses 
angular  momentum  be  left  to  itself,  it  cannot  move  as 
rigid  unless  the  axis  of  rotation  through  G,  the  centre  of 
mass,  be  fixed  in  the  body  and  coincide  with  the  axis  of 
resultant  angular  momentum  through  G,  which  is  a  line 
whose  directions  are  fixed  in  space.  Nevertheless  many 
bodies  are  so  hard  and  unyielding  that  we  may  often 
suppose,  without  serious  error,  that,  for  a  longer  or  shorter 
time,  they  move  as  rigid  whatever  be  their  axes  of  rotation. 
This  case  of  motion  alone  is  considered  in  books  on  Rigid 
Dynamics. 

We  have  seen  that  when  any  body  which  possesses 
angular  momentum  is  left  to  itself,  it  will  ultimately 
move  as  rigid  and  that  the  ultimate  axis  of  rotation 
through  the  centre  of  mass  will  be  fixed  in  the  body  and 
coincide  with  the  axis  of  resultant  angular  momentum 
through  G,  or  be  a  principal  axis  at  G.  This  is  true 


NOTE  B. 


383 


whether  the  body  be  hard  or  soft  ;  only  in  the  former 
case,  the  principal  axis  will  be  practically  ready-made, 
while  in  the  latter,  it  will  be  formed  to  a  greater  or  less 
extent  by  the  motion  of  the  body  itself.  But  since  we 
can  always  conceive  a  hard  and  nearly  rigid  body  having 
its  mass  distributed  in  exactly  the  same  way  as  any  given 
soft  body,  it  is  clear  that  a  principal  axis  always  exists 
ready-made  at  G,  however  soft  or  mobile  the  body  may  be. 
Suppose  Gz  to  be  a  principal  axis  at  G  and  let  it  be 
perpendicular  to  the  plane  of  the  paper.  Also  let  Gx,  Gy 
be  any  two  rectangular  axes  in  the  plane  of  the  paper. 
Then  if  (x,  y,  2)  be  the  coordinates  of  any  particle,  m, 
of  the  body,  with  respect  to  these  axes,  the  necessary  and 
sufficient  condition  that  Gz  should  be  a  principal  axis,  is 

^mzx  —  ^mzy  =  0. 
Now  if  Gx,  Gy'  be  any  other  pair  of  rectangular  axes  in 


the  plane  of  the  paper,  making  an  angle  6  with  the  first 
pair,  we  shall  have 

x  =  x  cos  6  +  y  sin  ff\ 

y'  =  y  cos  6  —  x  sin  6}  ' 


384 


ELEMENTARY  THERMODYNAMICS. 


and  therefore 

^mx'y  =  cos  20'Zmxy  -  sin  6  cos  02m  (x2  —  y2}. 
Thus  whatever   2mar?/  and  2m  (a?  —  y2)  may  be,  we   can 
always  choose  two  rectangular  axes  Gx ',   Gy',  such  that 
'Zmx'y'  =  0.     In   this   case,   the   three    rectangular    axes 
Gx',  Gy',  Gz  will  all  be  principal  axes. 

Now  let  Gz  be  at  any  instant  the  axis  of  rotation  and 
ft>  the  angular  velocity,  of  a  body  which  is  moving  as  rigid, 
and  let  Gz  be  any  other  axis  through  G  making  an  angle  6 
with  Gz.  Take  Gx,  Gx  in  the  plane  z'Gz,  at  right  angles 
to  Gz',  Gz,  respectively,  and  let  Gy'  be  perpendicular  to 
this  plane.  Then  since  the  angular  momentum  about  Gz' 
is  &)2m (x2  +  y'2\  and  about  Gx',  —  ay^mz'x,  the  angular 
momentum  about  Gz  will  be 

to  cos  02m  (x'2  +  y'2)  +  co  sin  Q^mz'x. 


But  since  z  =  x  sin  6  +  z  cos  &\ 

x  =  x  cos  6  —  z  sin  0}  ' 
this  becomes    eo  cos  #2w  (x2  cos2  6  +  z2  sin2  6  +  y'2) 

+  w  sin2  6  cos  02m  (it-2  -  z2) 
+  a)  {-  2  sin  6  cos2  0  +  sin  6  cos  2(9}  2m^, 
or  «  cos  02m  (#2  +  y'2)  —  w  sin  Q^mxz. 


NOTE    B.  385 


If  Gz  be  a  principal  axis  at  G,  ^inxz  =  0,  and  the  angular 
momentum  about  Gz  reduces  to  a>  cos  O^m  (a?  +  y'2).  This 
result  will  appear  to  be  very  simple  as  soon  as  we  have 
proved  that  angular  rotations  may  be  resolved  and 
compounded  like  forces,  so  that  to  cos  6  may  be  called  the 
resolved  angular  velocity  about  Gz. 

Suppose  that  angular  velocities  exist  in  succession  for 
the  same  short  time  r  about  two  straight  lines  OS,  OT, 
passing  through  a  fixed  point  0,  and  let  them  be  repre- 
sented in  magnitude  and  direction  by  the  lengths  OA  ,  OB. 
Complete  the  parallelogram  AOB,  and  from  any  point  P 
in  the  diagonal  OC  drop  perpendiculars  PM,  PN  on  OS 


M  A 


and  OT.  Then  an  angular  velocity  about  OS  proportional 
to  OA  will  cause  the  point  P  to  move  in  a  short  time  T 
perpendicular  to  the  plane  SOT  a  distance  proportional  to 
0 A  .  PM .  T.  If,  after  the  time  r,  the  angular  rotation 
about  OS  cease,  and  a  new  angular  velocity  proportional 
to  OB  begin  about  the  new  position  of  OT,  the  point  P 
will,  in  the  short  time  T,  be  brought  back  perpendicular 
to  the  plane  of  the  paper  a  distance  proportional  to 
OB  .  PN .  T.  Now  if  Cm,  Cn  be  the  perpendiculars  from 
C  on  OS  and  OT,  we  have  PM:  PN  =  Cm :  Cn,  and 
therefore  OA  .  PM  :  OB .  PN  =  OA  .  Cm  :  OB.  Cn=  1  : 1, 
since  the  triangles  CO  A,  COB  are  equal.  Hence  the 
point  P  is  brought  back  to  its  original  position  by  the 
P.  25 


386  ELEMENTARY   THERMODYNAMICS. 

second  rotation,  and  therefore,  since  any  displacement  of  a 
rigid  body  with  one  point,  0,  fixed,  can  be  effected  by  a 
rotation  about  an  axis  through  0,  the  two  rotations  OA, 
OB,  are  equivalent  to  a  single  rotation  about  the  diagonal 
OC,  or  angular  velocities  are  compounded  according  to  the 
parallelogramic  law. 

Again,  let  a  body  be  moving  as  rigid  and  take  three 
principal  axes  Gg ,  Gtj,  G£,  fixed  in  the  body,  as  rectangular 
axes  of  coordinates.  Let  the  moments  of  inertia  of  the 
body  about  these  axes  be  denoted  by  A,  B,  C,  which  will, 
of  course,  be  constant  quantities  however  the  axes  may 
move  about  in  space.  Then  if,  at  any  instant,  (a>1}  co2,  &>s) 
be  the  resolved  angular  velocities  about  these  axes, 
(Ao)l,  jB&>2,  €0)3)  will  be  the  resolved  angular  momenta. 
Hence  if  (a,  /3,  7)  be  the  angles  the  axis  of  rotation  makes 
with  the  axes  of  coordinates,  the  resolved  angular  mo- 
mentum about  the  axis  of  rotation  will  be 
Awl  cos  a  +  Bw»  cos  /3  +  Cco3  cos  7. 

To  get  the   mechanical   kinetic   energy  of  rotation,  we 
multiply  this  by  £&>.     It  is  therefore  equal  to 
1  (Aw?  +  Bay./  +  CM.?). 

If  t  be  the  time  at  which  the  angular  velocities  are 
(«!,  &>2,  »3)>  then  at  the  consecutive  instant  t  +  dt,  the  body 
will  have  angular  velocities  Wj  +  -,,1  dt,  &c.,  &c.,  about  the 
new  positions  of  Gg,  Grj,  G£  and  angular  momenta 
4  (<*i  +  d£dt},&C.,&c. 


If  we  denote  the  position  of  the  axes  at  the  time  t  by  G%, 
Gy,  G£,  and  at  the  time  t  +  dt,  by  £f ,  Grf,  G£,  the  angles 

between  G£  and  Grj',  G£,  will  be  practically  J  +  a)sdt  and 


NOTE   B. 


387 


2  -  w«dt     Hence  the  angular  momentum  of  the  body  at 
the  time  t  +  dt  about  G%  will  be 


so  that  if,  at  the  time  t,  L  be  the  moment  of  the  external 
forces  about  G£,  we  shall  have 

A  — j—  —  (Jj  —  C )  ft>.>&)3  =  LJ. 
at 

Similarly,  if  M,  N  be  the  moments  of  the  forces  about  the 
other  axes  at  the  time  t, 


and 


-(A -B) <*,<»,  =  N. 


25—2 


388  ELEMENTARY  THERMODYNAMICS. 

These  three  equations  are  due  to  Euler,  and  are  known 
as  Euler 's  equations. 

If  we  take  three  rectangular  axes  Ox,  Oy,  Oz,  fixed  in 
space,  and  denote  the  sums  of  the  resolved  parts  parallel 
to  these  axes  of  the  external  forces  which  act  on  the  body 
by  P,  Q,  R,  and  the  coordinates  of  G  by  (an,  y,  z\  we  also 
have 


where  M  is  the  mass  of  the  body. 

When  L,  M  and  N  are  zero,  we  obtain,  by  multiplying 
Euler's  equations  by  col,  w.2)  o>3,  respectively,  adding,  and 
integrating, 

\  (Awf  -t-  Bo).f  +  G'&v)  =  constant, 
or  the  mechanical  kinetic  energy  of  rotation  is  constant. 

When  P,  Q  and  R  are  zero,  we  find  in  a  similar 
manner, 


or  the  mechanical  kinetic  energy  of  translation  is  constant. 

When  there  are  no  external  forces  at  all,  the  total 
mechanical  kinetic  energy  is  constant. 

Now  we  have  seen  that  when  a  body  of  which  no  part 
of  the  energy  can  vary  but  the  mechanical  kinetic  energy, 
is  left  to  itself,  the  mechanical  kinetic  energy  continually 
decreases,  except  when  the  axis  of  rotation  is  a  principal 
axis.  The  results  which  have  been  just  obtained  cannot 


NOTE    B.  389 

therefore  be  strictly  true  except  when  the  axis  of  rotation 
is  a  principal  axis,  or  two  of  the  three  quantities  fa,  a>2,  a>3) 
zero.  Still  they  may  often  be  very  close  approximations 
for  a  short  time. 

To  find  the  mechanical  work  done  by  the  external 
forces,  we  take  account  only  of  the  mechanical  displace- 
ments of  the  points  of  application.  As  in  books  on  Rigid 
Dynamics,  we  shall  restrict  ourselves  to  the  case  in  which 
the  rods  &c.  by  which  the  external  forces  are  applied,  do 
not  slip  on  the  surface  of  the  body — a  very  simple  restric- 
tion which  is  usually  expressed  in  mysterious  unphysical 
language.  The  mechanical  displacement  of  the  point  of 
application  of  any  one  of  the  forces  will  then  be  the  same 
as  that  of  some  point  of  the  body  itself. 
Through  0  draw  three  fixed  rectangular  axes  Ox',  Oy',  Oz, 
parallel  to  the  moving  axes  Gg,  Gij,  G£  at  the  time  t. 
Then  if  (x',  y,  z'}  be  the  coordinates  of  G  at  the  time  t 
with  respect  to  these  axes,  and  (£,  rj,  f)  the  coordinates  of 
a  point  of  the  body  with  respect  to  the  moving  axes,  the 
mechanical  displacement  of  this  point  in  the  short  time  dt 
will  have  a  resolved  part  parallel  to  Ox  equal  to 

+(w£-wMdt. 

Hence  since  the  mechanical  work  done  by  a  force  is  equal 
to  the  sum  of  the  mechanical  works  done  by  its  com- 
ponents, if  (X1,  Y,  Z'}  be  the  resolved  parts,  parallel  to 
Ox',  Oy ,  Oz',  of  the  force  which  acts  at  (£,  77,  £),  the 
mechanical  work  which  this  force  does  in  the  time  dt  bears 
to  dt  the  ratio 

x>   ' + 


390  ELEMENTARY   THERMODYNAMICS. 

Thus,  since  2  (Z'rj  -  F£)  =  L 


the  mechanical  work  done  in  the  time  dt  by  all  the  forces 
will  be 

daf'S.X'  +  dy"Z  Y'  +  dz'^Z'  +  (Lm,  +  Mco,  +  No)s)  dt, 
which  is  clearly  equal  to 

Pdx  +  Qdy  +  Rdz  +  (Lfol  +  M(o.2  +  N<DS)  dt. 

Substituting  for  P,  Q,  E,  L,  M,  N,  their  values  as  already 
found,  the  expression  for  the  elementary  mechanical  work 
becomes 


^\drdt  + 

or  simply  dT, 

where  T  is  the  mechanical  kinetic  energy. 

Denoting  the   elementary  mechanical  work   by  dW,  we 

have 

dT=dW, 

and  we  see  that  the  mechanical  work  done  on  the  body  in 
any  finite  time  is  equal  to  the  corresponding  increase  of 
the  mechanical  kinetic  energy.  This  is  the  only  case  of 
the  principle  of  energy  which  is  considered  in  books  on 
Rigid  Dynamics  and  it  is,  in  general,  only  a  near  approxi- 
mation. 

This  result,  which  has  been  obtained  as  a  deduction 
from  the  laws  of  motion,  is  obvious  if  we  assume  the 
general  principle  of  energy.  For  since  the  only  part  of 
the  energy  of  the  body  which  can  vary  is  the  mechanical 
kinetic  energy,  the  increase  of  energy  in  any  short  interval 
is  dT,  and  since  the  only  kind  of  work  done  on  the  body 


NOTE   B.  391 

is  mechanical  work,  the  total  work  done  on  it  in  the  short 
interval  is  d  W.     Hence,  as  before,  dT  =  d  W. 

It  will  be  readily  understood  that  dW  need  not  be  a 
perfect  differential ;  nevertheless  the  impression  generally 
produced  by  the  ordinary  books  on  Rigid  Dynamics  is  that 
the  principle  of  energy  requires  that  d  W  should  always  be 
a  perfect  differential.  All  that  is  meant  is  the  following. 
The  given  body  is  supposed  to  form  part  of  one  system 
with  the  rods,  bars,  &c.,  and  external  gravitating  masses, 
to  which  the  external  forces  are  due.  The  new  system  is 
supposed  to  have  no  external  forces  acting  on  it,  nor  to 
receive  or  lose  heat,  and  it  is  further  supposed  that  the 
only  parts  of  its  energy  which  can  vary  are  the  mechanical 
kinetic  energy  of  the  given  body  and  the  mutual  potential 
energy,  V,  of  the  given  body  and  the  rest  of  the  system. 
The  principle  of  energy  then  gives 

T+  V=  constant, 

so  that  dW  is  the  complete  differential  of  the  function 
-F. 

As  we  have  already  said,  the  books  on  Rigid  Dynamics 
unfortunately  call  V  the  potential  energy  of  the  given 
body,  instead  of  the  mutual  potential  energy  of  the  given 
body  and  the  rest  of  the  system. 


APPENDIX. 


Densities  of  Gases, 

(In  the  last  two  columns,  the  pressure  is  supposed  to  be 
one  atmo  and  the  temperature  0°  C. ) 


Relative 
densities. 

Relative 

Mass  of  a 

Volume  of  a 

specific 
volumes. 

litre  in 
grammes. 

gramme  in 
litres. 

Air 

1 

1 

1-2932 

•7733 

Oxygen  (0) 

1-10563 

•90446 

1-4298 

•6994 

Hydrogen  (H) 
Nitrogen  (N) 

•06926 
•97135 

14-4383 
1-02945 

•08957 
1-25615 

11-16445 
•7961 

Carbonic  Oxide  (CO) 

•9545 

1-0476 

1-2344 

•8101 

Carbonic  Acid  (CO.,) 

1-52907 

•6540 

1-9774 

•5057 

Chlorine  (Cl) 

2-4222 

•4128 

3-1328 

•3192 

Cyanogen  (NC») 
Marsh  Gas  (Cfl.) 
Olefiant  Gas  (C2H4) 
Ammonia  (XH3) 

1-8019 
•562 
•982 
•5952 

•5550 
1-779 
1-018 
1-6801 

2-3302 
•727 
1-270 
•7697 

•4291 
1-375 

•787 
1-2992 

Specific  Heats  of  Gases. 


At  constant  pressure 

At  constant  volume 

In  calories. 
(Hv  experi- 
ment.) 

Compared 
with  an  equal 
vol.  of  air. 

Compared 
In  calories,    with  an  equal 
vol.  of  air. 

Ratio  of 
the  specific 
heats,  or  k. 

Air 

•2375 

1 

•1684          1 

•410 

Oxygen 
Hydrogen 

•21751 
3-40900 

1-012 
•994 

•15501 
2-4114 

1-018 
•992 

•403 
•414 

Nitrogen 

•24380 

•997 

•17266 

•996 

•412 

Carbonic  Oxide 

•2450 

•985 

•1728 

•978 

•418 

Carbonic  Acid 

•2169 

1-396 

•1717 

1-559 

•263 

Chlorine 

•12099 

1-234 

•0925 

1-330 

•308 

Marsh  Gas 

•5929 

1-403 

•4700 

1-568 

•260 

Olefiant  Gas 

•4040 

1-670 

•3337 

1-946 

1-211 

Ammonia 

•5084          1-274 

•3923          1-386          1-296 

394 


APPENDIX. 


Water  at  a  irressure  of  oiie  atmo. 

(Everett's  '  Units  and  Physical  Constants,'  also  '  Encyclopedia 
Britannica'.) 


Mass  of  one  cubic  centimetre  in 
grammes,  or  of  one  cubic  deci- 
metre in  kilogrammes. 

Volume  of  one  gramme  in  cubic  centi- 
metres, or  of  one  kilogramme  in 
cubic  decimetres. 

Temperature 

in  degrees 

Temperature. 

centigrade. 

0° 

•999,884 

0° 

1-000,116 

4° 

1-000,013 

4° 

•999,987 

5° 

1-000,003 

5° 

•999,997 

10° 

•999,760 

10° 

1-000,240 

15° 

•999,173 

15° 

1-000,828 

20° 

•998,272 

20° 

1-001,731 

25° 

•997,108 

25° 

1-002,900 

30° 

•995,778 

30° 

1-004,240 

35° 

•994,69 

35° 

1-005,34 

40° 

•992,36 

40° 

1-007,70 

45° 

•990,38 

45° 

1-009,71 

50° 

•988,21 

50° 

1-011,93 

55° 

•985,83 

55° 

1-014,37 

60° 

•983,39 

60° 

1-016,89 

65° 

•980,75 

65° 

1-019,63 

70° 

•977,95 

70° 

1-022,55 

75° 

•974,99 

75° 

1-025,65 

80° 

•971,95 

80° 

1-028,86 

85° 
90° 

•968,80 
•965,57 

85° 
90° 

1-032,20 
1-035,66 

95° 
100° 

•962,09 
•958,66 

95° 
100° 

1-039,40 
1-043,12 

According  to  Regnault,  the  specific  heat  of  water  (in  calo- 
ries) is  as  follows  : 

1-0000 


at  0°  C. 
at  10°  „ 
at  20°  „ 
at  30°  „ 


1-0005 
1-0012 
1-0020 


APPENDIX. 


395 


Ice  at  the,  pressure  of  one  atmo  and  at  0°  C. 

Mass  of  one  cubic  centimetre  =  '920  gramme. 
Volume  of  one  gramme  =  1-087  cubic  centimetres. 
Coefficient  of  cubical  dilatation  by  heat  at  constant  pres- 
sure =  -000153. 
Specific  heat  (in  calories)  at  constant  pressure  ---  '48. 

(Person.) 

Mercury  at  a  pressure  of  one  atmo  (or  less). 

It  has  been  found  (Everett's  'Units')  that  the  density  of 
mercury  at  0°  C.  is  13-5956  times  that  of  water  at  4°  C.  It  is 
therefore  13*595  776  74  grammes  per  cubic  centimetre.  Hence 
the  pressure  produced  (at  Paris)  by  a  column  of  mercury  at 
0°  C.  and  one  millimetre  high,  whose  top  is  acted  on  by  no 
force  but  the  insignificant  pressure  of  its  own  saturated  vapour, 
is  1333-5662  dynes  per  square  centimetre.  Also  an  atmo,  or 
the  pressure  produced  (at  Paris)  by  a  column  of  mercury  at 
0°  C.  and  760  millimetres  high,  is  1,013,510-3356  dynes,  or 
1033-279  grammes,  per  square  centimetre. 

Melting  Points  and  Latent  Heats  of  Fusion  (in  calories) 

of  Solids  at  a  pressure  of  one  atmo. 
(From  Watt's  'Dictionary  of  Chemistry.') 


Melting-points  (C.).        Latent  Heats. 

Mercury 
Phosphorus 
Sulphur 

-39° 
44°-2 
115° 

2-82 
5-0 
9-4 

Iodine 

107° 

11-7 

Lead 

332° 

5-4 

Tin 

235° 

14-25 

Silver 

1000° 

2M 

Zinc 

433° 

28-1 

Bismuth 

270° 

12-6 

Nitrate  of  Potassium 

339° 

47-4 

Nitrate  of  Sodium                     310°  -5 

63-0 

?6  APPENDIX. 

Roiling  Points  and  Heats  of  Vaporisation  at  a  pressure 
of  one  atmo. 

(Everett's  'Units'.) 


Boiling-points  (C). 

Latent  Heat  of 
Vaporisation. 

Observer. 

Alcohol 

77°-9 

202-4 

Andrews 

Bisulphide  of  ) 

o 

Carbon          J 

46  '2 

86'7 

» 

Bromine 

58° 

45-6 

M 

Ether 

34°-9 

90-4 

}j 

Mercury 

350°  (?) 

62 

Person 

Sulphur  . 

316°  (?) 

362 

„ 

also 

Sulphurous  Acid 

-  103-08 

Concentrated      } 

Sulphuric  Acid  ] 

325° 

Temperatures  and  Pressures  of  Critical  Points. 
(Cagniard  de  la  Tour.) 


|  Critical  Temperature. 

Pressure  in  atmos. 

Bisulphide  of  Carbon 
Ether 
Alcohol 
Water 

262-5°  C. 
187-5°  C. 
258-7°  C. 
411-7°  C. 

66-5 
37-5 
119 
? 

In  the  case  of  water,  Maxwell  estimates  the  Critical  Tem- 
perature to  be  about  434°  C.,  the  Critical  Pressure  about 
378  atmos,  and  the  Critical  Volume  about  2-52  cubic  centi- 
metres per  gramme. 

Pressure  of  ilie  saturated  vapour  of  Water 
(calculated  from  Eegnault). 

[When  the  pressure  is  measured  by  a  column  of  mercury, 
the  mercury  is  supposed  to  be  contained  in  a  wide  (non- 


APPENDIX. 


397 


capillary)  tube  at  Paris,  at  0°  C.,  and  subjected  to  no  surface 
pressure  beyond  the  negligible  pressure  of  its  own  vapour.] 

Pressures. 


Tempera- 
tures (C.). 

In  millimetres 
of  mercury. 

1  11  grammes  (at 
Paris)  per  square 
centimetre. 

In  dynes 
per  square 
centimetre. 

In  pounds 
In  atmos.      iper  sq.  inch 
(atLondon). 

-30° 

•4 

•54 

529-7 

•008 

-25° 

•6 

•82 

804-3 

•012 

-20° 

•9 

1-22                 1,196-7 

•001              -017 

-15° 

1-4 

1-90                1,863-6 

•002              -027 

-10° 

2-1 

2-85                2,795-5 

•003 

•041 

-    5° 

3-1 

4-21                 4,129-4 

•004 

•059 

0° 

4-600 

6-254     ;          6,134-40 

•006 

•089 

5° 

6-534 

8-883              8,713-52 

•0086 

•126 

10° 

9-165 

12-460 

12,222-1 

•0120 

•177 

15° 

12-699 

17-265 

16,934-9 

•0168     I       -246 

20° 

17-391 

23-H44 

23,192-0 

•0229            -336 

25° 

23-550 

32-018     ;        31,405-5 

•0309     !       '455 

30° 

31-548 

42-892     i        42,071-3 

•0415     ;       -610 

35° 

41-827 

56-867     j        55,779-1 

•0550     :       "809 

40° 

54-906 

74-649            73,220-8 

•0722          1-062 

45° 

71-390 

97-060            95,203-3 

•0939         1-380 

50° 

91-980 

125-054          122,661 

•1210 

1-779 

55° 

117-475 

159-716          156,661 

•1545 

2-27 

60° 

148-786 

202-262 

198,416 

•1957 

2-88 

65° 

186-938 

254-247 

249,294 

•246 

3-61 

70° 

233-082 

316-893 

310,830 

•306 

4-51 

75° 

288-500 

392-238 

384,734 

•379 

5-58 

80° 

354-616 

482-128 

472,904 

•467 

6-86 

85° 

433-002 

588-700 

577,437 

•570 

8-37 

90° 

525-392 

714-311 

700,645 

•704 

10-16 

95° 

633-692 

861-553 

845,070 

•833 

12-25 

100° 

760              1033-279 

1,013,510 

1 

14-697 

105° 

906-41          1232-33 

1,208,760 

1-19 

17-53 

110° 

1075-37 

1462-05 

1,434,080 

1-41 

20-80 

115° 

1269-41 

1725-86 

1,692,840 

1-67 

24-55 

120° 

1491-28     |     2027-51 

1,988,720 

1-96 

28-84 

125° 

1743-88 

2370-94 

2,325,580 

2-29 

33-72 

130° 

2030-28 

2760-32 

2,707,510 

2-67 

39-26 

135° 

2353-73 

3200-08 

3,138,850 

3-09 

45-5 

140° 

2717-63 

3694-83 

3,624,140 

3-57 

52-6 

145° 

3125-55 

4249-43 

4,168,130 

4-11 

60-4 

150° 

3581-23 

4868-97 

4,775,810 

4-71 

69-3 

155° 

4088-56 

5558-71 

5,452,370             5-38 

79-1 

398 


APPENDIX. 


Tempera- 
tures (C.). 

]n  millimetres 
of  mercury. 

In  grammes  (at 
Paris)  pi'rsc[viiirc 
centimetre. 

In  dynes 
per  square 
centimetre. 

In  atmos. 

In  pounds 
per  sq.  inch 
(atLondon). 

160° 

4651-62 

6324-24 

6,203,240 

6-12 

90-0 

165° 

5274-54 

7171-15 

7,033,950 

6-94 

102-0 

170° 

5961-66 

8105-34 

7,950,270 

7-84 

115-3 

175° 

6717-43 

9132-87 

8,958,140 

8-84 

129-9 

180° 

7546-39 

10259-9 

10,063,600 

9-93 

145-9 

185° 

8453-23 

11492-8 

11,272,900 

11-12 

163-5 

190° 

9442-70 

12837-1 

12,592,500 

12-42 

182-6 

195° 

10519-63 

14302-2 

14,028,600 

13-84 

203-4 

200° 

11688-96 

15892-0 

15,588,000 

15-38 

226-0 

205° 

12955-7 

17614-3 

17,277,300 

17-0 

250-5 

210° 

14324-8 

19475-7 

19,103,100 

18-8 

277-0 

215° 

15801-3 

21483-1 

21,072,100 

20-8 

305-6 

220° 

17390-4 

23643-6 

23,191,200 

22-9 

336-3 

225° 

19097 

25963-8 

22,800,000 

25-1 

369-3 

230° 

20926-4       I     28451-1 

27,906,700 

27'5 

404-7 

Pressure  of  saturated  vapour  of  Mercury  (in  millimetres 
of  mercury}. 

(From  the  'Encjclop.  Brit.') 


Temperatures 
(C.). 

Pressures. 

Temperatures 

Pressures. 

Temperatures 

Pressures. 

0° 

•02 

180° 

11 

360° 

7977 

10° 

•03 

190° 

14-8 

370° 

954-6 

20° 

•04 

200° 

19-9 

380° 

1139-6 

30° 

•05 

210° 

26-3 

390° 

1346-7 

40° 

•08 

220° 

34-7 

400° 

1588 

50° 

•11 

230° 

45-3 

410° 

1863-7 

60° 

•16 

240° 

58-8 

420° 

2177-5 

70° 

•24 

250° 

75-7 

430° 

2533 

80° 

•35 

260° 

96-7 

440° 

2934 

90° 

•51 

270° 

123 

450° 

3384-4 

100° 
110° 

•75 
1-07 

280° 
290° 

155-2 
194-5 

460° 
470° 

3888-1 
4449-4 

120° 

1-53 

300° 

242-2 

480° 

5072-4 

130° 
140° 
150° 
160° 
170° 

2-18 
3-06 
4-27 
5-90 
8-09 

310° 
320° 
330° 
340° 
350° 

299-7 
368-7 
450-9 
548-3 
663-2 

490° 
500° 
510° 

520° 

5761-3 
6520-3 
7253-4 
8265 

APPENDIX. 


399 


Pressure  of  saturated  vapour  of  Sulphur  (in  millimetres 

of  mercury). 
('Encyclop.  Brit.') 


Temps.    (C.) 

Pressures.      Temps.    (C.)  ,    Pressure. 

Temps.    (C.) 

Pressures. 

390° 

i     272-3           460°             912-7 

520° 

2133-3 

400° 

329              470°            1063-2 

530° 

2422 

410° 

395-2           480°            1232-7 

540° 

2739-2 

420° 

472-1            490°            1422-9 

550° 

3086-5 

430° 

561               500°            1635-3 

560° 

3465-3 

440° 

663-1            510°            1871-6 

570° 

3877-1 

450° 

779-9 

Pressures  of  saturated  vapours  (in  millimetres  of  mercuri/). 

('Encyclop.  Brit.') 

Temps. 

Ammonia. 
(NH3). 

Sulphuretted 
Hydrogen.   (H2S). 

Jarbonic  Acid.         Nitrous  Oxide. 
(C02).                        (XoO). 

-30° 

866-1 

-25° 

1104-3 

3749-3 

13007                 15694-9 

-20° 

1392-1 

4438-5 

15142-4             17586-6 

-15° 

1736-5 

5196-5 

17582-5              19684-3 

-10° 

2144-6 

6084-6 

20340-2 

22008 

-    5° 

2624-2 

7066 

23441-3 

24579-2 

0° 

3183-3 

8206-3 

26906-6 

27421 

5° 

3830-3 

9490-8 

30753-8 

30558-6 

10° 

4574 

10896-3 

34998-6             34019-1 

15° 

5423-4 

12447-9 

39646-9             37831-7 

20° 

6387-8 

14151-5 

44716-6 

42027-9 

25° 

7477 

16012-4 

50207-3 

46641-4 

30° 

8701 

18035-3 

56119 

51708-5 

35° 

10070-2 

20224-3 

62447-3 

57268-1 

40° 

11595-3 

22582-5 

69184-4 

63359-8 

45° 

13287-3 

24954-3 

76314-6 

50° 

15158-3 

27814-8 

55° 

17219-8             30690-7 

60° 

19482-1              33740-2 

65° 

21965-1             36961-5 

70° 

24675-5              40353-2 

75° 

27630 

80° 

30843-1 

85° 

34330-9 

90° 

38109-2 

95° 

42195-7 

100° 

46608-2 

400 


APPENDIX. 


Pressures  of  saturated  vapours  (in  millimetres  of  mercury). 

('Encyclop.  Brit.') 


Temps. 
(C). 

Essence  of 
Turpentine. 
(t'ioH6). 

Chloroform. 

(Ciicy. 

Carbon 
Bisulphide.  (CS.,). 

Sulphurous 
Acid.  (SO-i). 

-30° 

287-5 

-25° 

373-8 

-20° 

47-3 

479-5 

-15° 

61-6 

607-9 

-  10° 

79-4 

762-5 

-    5° 

101-3 

946-9 

0°                2-1 

127-9 

1165-1 

5° 

160 

1421-1 

10°                2-9 

198-5 

1719-5 

15° 

244-1 

2064-9 

20° 

4-4 

160-5 

298 

2462 

25° 

200-2               361-1 

2916 

30°                6-9 

247-5                434-6 

3431-8 

35° 

303-5 

519-7 

4014-8 

40° 

10-8 

369-3 

617-5 

4670-2 

45° 

446 

729-5 

5403-5 

50° 

17 

535 

857-1 

6220 

55° 

637-7 

1001-6 

7125 

60° 

26-5 

755-4 

1164-5 

8123-8 

65° 

889-7 

1347-5 

9221-4 

70° 

40-6 

1042-1 

1552-1 

75° 

1214-2 

1779-9 

80°             61-3 

1407-6 

2032-5 

85° 

1624-1 

2311-7 

90° 

90-6 

1865-2 

2619-1 

95° 

2132-8              2966-3 

100° 

131-1 

2428-5              3325-1 

105° 

2754 

3727-2 

110° 

186 

3111 

4164-1 

115° 

3501 

4637-4 

120°           257 

3925-7 

5148-8 

125° 
130°            349 

4386-6 
4885-1 

5699-7 
6291-6 

135° 
140° 

464 

5422-5 
6000-2 

6925-9 
7604 

145° 
150° 
155° 

605 
686 

6619-2 
7280-6 
7985-3 

8326-9 
9095-9 

160° 

775 

8734-2 

165° 

9527-8 

APPENDIX.  401 

Solution  of  Gases  in  Water. 
(From  Koscoe  and  Schorlemmer's  'Chemistry'.) 

The  number  of  volumes  of  gas  (reduced  to  0°  C.  and  one 
atmo)  absorbed  by  one  volume  of  water,  is  called  the  'coefficient 
of  solubility  '  of  the  gas. 

Oxygen  (0). 

The  coefficient  of  solubility  (7,  at  the  temperature  0°C.,  is 
given  by 

0  =  -04115  -  -0010899  6  +  -000022563  6\ 

Hydrogen  (H). 

Unlike  most  other  gases,  Hydrogen  is  equally  soluble  for 
all  temperatures  between  0°C.  and  20°  C.,  the  coefficient  of 
solubility  being  -0193. 

Nitrogen  (N). 
The  coefficient  of  solubility  is 

C  =  -020346  -  -00053887  0  +  '00001156  62. 

Sulphuretted  Hydrogen  (H2S). 

Between  O'C.  and  40°  C.,  the  coefficient  of  solubility  is 
C  =  4-3706  -  -0836870  +  -000521302. 

Carbonic  Acid  (CO2). 

Specific  gravity  of  liquid  Carbonic  Acid  is  -9951  at  -10°  C., 
•9470  at  0°C.,  and  -8266  at  20° C.  It  is  thus  very  expansible 
by  heat. 

The  boiling  point  under  a  pressure  of  one  atmo  is  -78-2°C. 
For  the  coefficient  of  solubility  we  have  the  formula 

P.  26 


402  APPENDIX. 

c  =  i-7967  -  -07761 6  +  -001642402, 
or 

at    0°,  (7=1-7967 

„    5°,  (7=1-4497 

„  10°,  (7=1-1848  (•- 

„  15°,  (7=1-0020 

„  20°,  (7=    -9014 

When  the  pressure  is  much  smaller  than  that  of  the  atmo- 
sphere, the  quantity  of  gas  absorbed  at  a  given  temperature  is 
proportional  to  the  pressure. 

Chlorine  (01). 

The  coefficient  of  solubility  between  10°  C.  and  40°  C.  is 
given  by  the  formula 

C  =  3-0361  -  -0461960  +  -0001107  O2. 
Hence 

at  10°,  (7  =  2-5852 
„  20°,  (7  =  2-1565 
„  30°,  (7=1-7499 
„  40°,  (7  =  1-3654 

Ammonia  (NH3). 

The  coefficient  of  dilatation  by  heat  of  liquid  ammonia  at 
0°C.  is  -00204:  the  specific  gravity  at  the  same  temperature 
compared  with  water  is  "6234. 

It  has  been  found  by  Roscoe  and  Dittmar  that  the  quantity 
of  ammonia  absorbed  at  a  given  temperature  is  not  proportional 
to  the  pressure;  but  the  deviations  diminish  as  the  temperature 
increases. 

The  solubility  of  ammonia  at  different  temperatures  under 
a  pressure  of  one  atmo  is  shown  in  the  following  table,  due  to 
Roscoe  and  Dittmar. 


APPENDIX. 


403 


Temps.     (C.) 

Grammes  of  gas 
absorbed  bv  one 
gramme  of  water. 

Temps.    (C.) 

Grammes  of  gas 
absorbed  by  one 
gramme  of  water. 

0° 

•875 

30° 

•403 

2° 

•833 

32° 

•383 

4° 

•792 

34° 

•362 

6° 

•751 

36° 

•343 

8° 

•713 

38° 

•324 

10° 

•679 

40° 

•307 

12° 

•645 

42° 

•290 

14° 

•612 

44° 

•275 

16° 

•582 

46° 

•259 

18° 

•554 

48° 

•244 

20° 

•526 

50° 

•229 

22° 

•499 

52°                     -214 

24° 

•474 

54° 

•200 

26° 

•449 

56° 

•186 

28° 

•426 

Hydrochloric  Acid  (HC1). 

Hydrochloric  Acid  is  very  soluble  in  water,  and  the  solu- 
tion is  frequently  called  '  muriatic  acid.'  The  quantity  of  gas 
dissolved  at  a  given  temperature  is  not  proportional  to  the 
pressure. 

The  mass  of  hydrochloric  acid  absorbed  under  the  pressure 
of  one  atmo  by  one  gramme  of  water  at  different  temperatures 
is  given  by  the  following  table. 


Temps.   (C.) 

Grammes  of  HC1. 

Temps.    (C.) 

Grammes  of  HC1. 

0° 

•825 

32°                   -665 

4° 

•804 

36°                    -649 

8° 

•783 

40°                    '633 

12° 

•762 

44° 

•618 

16° 

•742 

48° 

•603 

20° 

•721 

52° 

•589 

24° 

•700 

56° 

•575 

28° 

•682 

60°                    -561 

404 


APPEXDIX. 


Dilution  of  Sulphuric  Acid  (H2S04)  with  water. 

When  Sulphuric  Acid  is  mixed  with  water,  there  is  a  con- 
traction of  volume  and  a  considerable  evolution  of  heat.  The 
quantity  of  heat  that  must  be  carried  off  to  keep  the  tempera- 
ture constant  has  been  determined  by  Thomsen,  from  whose 
results  the  following  table  is  derived.  We  suppose  that  there 
is  originally  one  gramme  of  (HC1)  and  that  a  solution  is  formed 
by  adding  x  molecules  of  (H20)  to  each  molecule  of  (HC1), 

that  is,  by  adding    ^  ,   grammes  of  (H,0). 


Values 
of  a-. 

Heat  evolved 
(in  calories). 

Values                    Heat 
of  j\                   evolved. 

1 

64 

49                 170 

2 

96 

99 

172 

3 

113 

199 

174 

5 

133 

499 

175 

9 

152 

799 

180 

19                 166 

1599 

182 

Solution  of  Salts  in  water. 

The  accompanying  table  exhibits  some  fundamental  proper- 
ties of  salts : 

(1)  The  amount  of  heat  that  must  be  imparted  to  keep 
the  temperature  constant  when  one  gramme  of  salt  is  thrown 
into  a  large   quantity  of   water.     (Favre   and   Silberman  in 
'  Watt's  Dictionary  of  Chemistry.') 

(2)  The  Specific  Heats  of  salts.     (Regnault  in  'Watt's 
Dictionary  of  Chemistry.') 

(3)  The   approximate   Specific   Gravities   from    'Clarke's 
Constants  of  Nature.' 


APPENDIX. 


405 


Name  of  Salt. 

Heat 
absorbed. 

Specific 
heat 

Specific 
gravity. 

Sulphate  of    Potassium 

35-3 

2-6 

„         ,     Sodium 

49-1 

2-65 

„         ,     Calcium 

24-7 

•19656 

„         ,     Ammonium 

11-1 

„         ,     Zinc 

14-8 

„         ,     Lead 

•08723 

6-25 

Chloride  of    Sodium 

8-9 

•21401 

2-15 

„               Calcium  (CaCl2) 

15-5 

•16420 

2-25 

„               Potassium 

51-9 

•17295 

1-98 

„               Magnesium  (HgCl.,) 

•19460 

„                Zinc 

•13618 

Lead 

•06641 

Nitrate  of      Sodium 

45-5 

•27821 

2-25 

„        „       Calcium 

27-1 

„        „       Potassium 
Carbonate  of  Sodium 

52-7 

•23875 
•23115 

2 
2-45 

,,          „  Potassium 

•19010 

2-21 

Salts  may  be  divided  into  two  classes :  (1)  those  whose 
solution  deposits  the  anhydrous  salt ;  (2)  those  which  deposit 
a  hydrate. 

To  the  first  class  belong  : — 

Sulphate  of  Ammonium 
Chlorate  of  Potassium 
Chloride  of  Ammonium 
Nitrate  of  Potassium 
„       „  Lead 
„       „   Silver 
Iodide  of  Potassium 

To  the  second  class  belong  : — 

Sulphate  of  Ammonium 
„          „    Magnesium 

„    Zinc 

Chloride  of  Sodium 
.    Barium 


406 


APPENDIX. 


A  saturated  saline  solution  always  boils  at  a  temperature 
above  100°C.  But  the  excess  of  the  boiling  point  above 
100°  C.  is  not  proportional  to  the  amount  of  salt  dissolved. 
Thus 


Boiling-point. 

100  parts  of  water 
dissolve. 

Chloride  of  Sodium 
Nitrate  of  Potassium 
Carbonate  of  Potassium 
Iodide  of  Potassium 

109°  C. 
114°C. 
135°  C. 
118°  G. 

38-7 
327-4 
205-0 
223-0 

The  vapour-pressure  of  an  aqueous  solution  of  a  salt  is 
always  less  than  the  pressure  of  saturated  steam  at  the  same 
temperature,  the  difference  at  any  given  temperature  being 
roughly  proportional  to  the  percentage  of  salt  in  the  solution. 
Thus  at  51 -8° C.,  the  vapour-pressures  of  aqueous  solutions  of 
chloride  of  sodium,  expressed  in  millimetres  of  mercuiy,  were 
found  by  Wullner  to  be  :— 

(II20)  (NaCl)  millims. 

100  0  100 

90  10  94 

80  20  88 

Wullner  has  also  found  that  the  diminution  of  the  vapour- 
pressure  due  to  the  presence  of  a  given  percentage  of  any  salt 
increases  with  the  temperature. 

Table  shewing  the  temperatures  at  which  a  solid  first 
appears  on  cooling  solutions  of  chloride  of  sodium  of  different 
strengths.  (Dr  Guthrie  in  Phil  Mag.,  May,  1876.) 


NaCl  per  cent, 
by  weight. 

IIoO  per  cent, 
by  weight. 

Temperature  (C.)  at 
which  a  solid  formed. 

Nature  of 
Solid. 

1 

99 

-      -3° 

Ice 

2 

98 

-      -9° 

3 

97 

-    1-5° 

4                    96 

-    2-2° 

7            1         93 

-    4-2° 

APPENDIX. 
Table  (continued). 


407 


NaCl  per  cent        H50  per  cent.       Temperature  (C.)  at 
by  weight.              by  weight.         which  a  solid  formed. 

Nature  of 
Solid. 

10 

90                     -   6-6° 

Ice 

13 

87                     -   9-1° 

J, 

15 

85                      -11° 

16 

84 

-11-9° 

M 

19 

81 

-  15-5° 

20 

80 

-  17° 

22 

78 

-20° 

23-6 

76-4 

-22° 

Crvohvdrate 

25 

75 

-  12°                 Bihydrate 

(NaCl+2H20) 

26-27 
26-5 

73-73 
73-5 

0° 

+  25° 

5) 

26-8 


+  40° 


Some  of  the  principal  results  of  Dr  Guthrie's  experiments 
on  the  freezing  of  saline  solutions  are  given  below;  showing 
(1)  the  lowest  temperature  which  he  obtained  by  means  of  the 
cryogen,  (2)  the  freezing  point  of  the  cryohydrate  (under  a 
pressure  of  one  atmo),  (3)  the  'water-worth'  of  the  cryo- 
hydrate— that  is,  the  number  of  molecules  of  water  to  each 
molecule  of  the  salt.  (Phil.  Mag.,  April,  1875.) 


Salt. 

Temp,  of 
cryogen 

Temp,  of 
solidification 
of  cryohydrate 
(C.) 

Water- 
worth  of 

hydrate. 

Bromide  of  Sodium  (NaBr) 

-28° 

-24° 

8-1 

Iodide  of  Ammonium  (NH4I) 

-27° 

-  27-5° 

6-4 

Iodide  of  Sodium  (Xal) 
Iodide  of  Potassium  (KI) 

-  26-5° 
-22° 

-22° 

5-8 
8-5 

Chloride  of  Sodium  (NaCl) 

-22° 

-22° 

10-5 

Chloride  of  Strontium  (SrCl2-(-6H20) 

-18° 

-17° 

22-9 

Sulphate  of  Ammonium  (NlI4S04) 
Bromide  of  Ammonium  (NH4Br) 

-17-5°        -17C 
-17°           -17° 

10-2 
11-1 

Nitrate  of  Ammonium  (NH4N03) 

-17°           -17-2° 

5-72 

Nitrate  of  Sodium  (NaNOO 

-16-5°        -17-5° 

8-13 

Chloride  of  Ammonium  (NH4C1) 

-16°           -15° 

12-4 

Bromide  of  Potassium  (KBr) 

-13°           -13° 

13-94 

408                                              APPENDIX. 

Table  (continued). 

Temp,  of 

Temp,  of         Water- 
solidification     worth  of 

Salt. 

(C.) 

of  cryohvdrate       cryo- 
(C.)            hydrate. 

Chloride  of  Potassium  (K  Cl) 

-  10-5° 

-11-4° 

16-61 

Chromate  of  Potassium  (K.,Cr04) 

-  10-2° 

-12° 

18-8 

Chloride  of  Barium  (BaCL>  +  2H,0) 

-    7-2° 

-    8° 

37-8 

Nitrate  of  Lead  {Pb2(NO,)> 

-    2-5° 

-    2-5° 

Nitrate  of  Strontium  (Sr2N03) 

-    6° 

-   6° 

33-5 

Sulphate  of  Magnesium  (~MgS04  +  7H9O) 
Sulphate  of  Zinc  (ZnSO4  +  7H90) 
Nitrate  of  Potassium  (KNO.)  " 

-    5-3° 
-    5° 
-   3° 

-   5° 
-   2-6° 

23-8 
20 
44-6 

(Na2C03) 

-   2-2° 

-    2° 

92-75 

Sulphate  of  Copper  (CuS04  +  5H,0) 

-    2° 

-    2° 

43-7 

Sulphate  of  Iron  (FeS04  +  7H20)~ 
Sulphate  of  Potassium  (K2S04) 
Bichromate  of  Potassium  (K9CrO-) 
Nitrate  of  Barium  {Ba2(N03)} 

-    1-7° 
-    1-5° 
-    1° 
-      -9° 

-   2-2° 
-    1-2° 
-    1° 

-      -8° 

41-41 
114-2 
292 
259 

Crystallized  Sulphate  of  Soda  (Na.,S04 

+  10H,0) 

—     '7° 

-     -7° 

165-6 

Chlorate  of  Potassium  (KC10S) 

—     '7° 

-      -5° 

222 

Ammonium  Alum  {A12NH42(S04) 

+  12H20} 

-      -4° 

-      -2° 

261-4 

Perchloride  of  Mercury  (HgCl,) 
Nitrate  of  Silver  (AgN03) 
Anhydrous  Sulphate  of  Soda  (Na2S04) 

-      -2° 
-    6-5° 

-      -2° 
-    6-5° 

450 
10-09 

Anhydrous  Sulphate  of  Copper  (CuS04)    ! 
Nitrate  of  Calcium  {Ca2(NO,)j 
Chloride  of  Calcium  (CaCl2  +  3H20) 

-    1-7° 
-16° 

-37° 

11-8 

CAMBRIDGE:  PRINTED  BY  C.  J.  CLAY,  M.A.  AND  SONS,  AT  THE  UNIVERSITY  PRESS. 


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